EM diagnostics

probe the properties of spacetime in the strong-field region and may lead to constraining the nature of the compact object. We focus our discussion on three popular EM probes, namely shadows, X-ray spectra, and quasi-periodic oscillations. We do not discuss here polarization, nor effects on stellar trajectories. Because all EM tests can be eventually traced back to geodesic motion, EM probes may distinguish between BHs and ECOs with M/R < 0.492, whereas it is much more challenging to tell a ClePhO from a BH through EM measurements.

Shadows BHs appear on a bright background as a dark patch on the sky, due to

photons captured by the horizon. This feature is known as the BH shadow [1697,1698]. The Event Horizon Telescope [1699], aiming at obtaining sub-millimeter images of the shadow of the supermassive object Sgr A* at the center of the Milky Way and of the supermassive object at the center of the elliptical galaxy M87, is a strong motivation for these studies.

A rigorous and more technical definition of the BH shadow is the following [1700]: it is the set of directions on an observer’s local sky that asymptotically approach the event horizon when photons are ray traced backwards in time along them. Thus, by this definition, shadows are intrinsically linked with the existence of a horizon and, strictly speaking, an ECO cannot have a shadow. However, in practice ultracompact horizonless objects (in particular UCOs and ClePhOs) might be very efficient in mimicking the exterior spacetime of a Kerr BH. It is therefore interesting to study photon trajectories in such spacetimes and to contrast them with those occurring in the Kerr case.

In the analysis of shadows, one generally either considers parametrized spacetimes [1570, 1701–1703] (that allow to tune the departure from Kerr but might not map to known solutions of some theory), or takes into account a specific alternative theoretical framework [1704–1711] or a particular compact object within GR [1712– 1715]. Most of these studies obtain differences with respect to the standard Kerr spacetime that are smaller than the current instrumental resolution (< 10 µas). Some studies report more noticeable differences in the case of naked singularities [1712], exotic matter that violates some energy condition [1713], or in models that allow for large values of the non-Kerrness parameters [1708,1709]. Moreover, demonstrating a clear difference in the shadow is not enough to infer that a test can be made, since such difference might

be degenerate with the mass, spin, distance, and inclination of the source. Finally, the fact that horizonless objects lead to shadow-like regions that can share some clear resemblance with Kerr [1715] shows the extreme difficulty of an unambiguous test based on such observables (for perspective tests with current observations see, e.g., Refs. [1716,1717]).

Kα iron line and continuum-fitting The X-ray spectra of BHs in X-ray binaries and

AGNs are routinely used to constrain the spin parameter of BHs, assuming a Kerr metric [1718, 1719]. In particular, the iron Kα emission line, that is the strongest observed and is strongly affected by relativistic effects, is an interesting probe.

Many authors have recently investigated the X-ray spectral observables associated to non-Kerr spacetimes [1720]. The same distinction presented above for the tests of shadows can be made between parametric studies [1721–1723], specific alternative theoretical frameworks [1707,1722, 1724–1726], and alternative objects within classical GR [1727,1728].

Although differences with respect to Kerr are commonly found in various frameworks, these are generally degenerate with other parameters, such as mass, spin and inclination of the source. Moreover, the precise shape of the iron line depends on the subtle radiative transfer in the accretion disk, which is ignored in theoretical studies that generally favor simple analytic models. This is a great advantage of the shadow method, which is probably the EM probe least affected by astrophysical systematics.

QPOs Quasi-periodic oscillations (QPOs) are narrow peaks in the power spectra, routinely observed in the X-ray light curves of binaries. QPOs are of the order of Keplerian frequencies in the innermost regions of the accretion flow [1729].

A series of recent works were devoted to studying the QPO observables associated to alternative compact objects, be it in the context of parametric spacetimes [1730,1731], alternative theoretical frameworks [1707, 1732, 1733], or alternative objects within GR [1734]. Although the QPO frequencies can be measured with great accuracy, QPO diagnostics suffer from the same limitations as the X-ray spectrum: degeneracies and astrophysical uncertainties. The non-Kerr parameters are often degenerate with the object’s spin. Moreover, it is currently not even clear what the correct model for QPOs is.

Pulsar-BH systems High-precision timing observations of radio pulsars provide very

sensitive probes of the spacetime in the vicinity of compact objects. Indeed, the first evidence for the existence of gravitational waves were obtained from observations of binary pulsars [1735]; light propagation in strong gravitational effects can be precisely tested with Shapiro delay experiments [1736]; relativistic spin-precession can be studied by examining pulse shape and polarisation properties of pulsars [1737, 1738]. The very same techniques can be applied for pulsars to be found around BHs [1739].

While shadows, accretion-disk spectra, and QPOs are probing the near field of the BH, i.e. on a scale of a few Schwarzschild radii, it is not expected to find a pulsar that close to a BH. The lifetime of such a pulsar-BH system due to gravitational wave damping is very small, which makes a discovery of such a system extremely unlikely. This is also true for a pulsar around Sgr A∗, although observational evidence stills point towards an observable pulsar population in the Galactic Centre[1740]. Consequently, a pulsar-BH system, once discovered, is expected to provide a far field test, i.e. a test of only the leading multipole moments of the BH spacetime, in particular mass M•, spin

S• and — for IMBHs and Sgr A∗ — the quadrupole moment Q• [1741–1743]. On the

one hand, the measurement of M•, S• and Q• can be used to test the Kerr hypothesis.

On the other hand, a pulsar can provide a complementary test to the near field test, and break potential degeneracies of, for instance, a test based on the shadow of the BH[1702,1717, 1744].

A pulsar in a sufficiently tight orbit (period . few days) about a stellar-mass BH is expected to show a measurable amount of orbital shrinkage due to the emission of gravitational waves. Alternatives to GR generally show the existence of “gravitational charges” associated with additional long-range fields, like e.g. the scalar field in scalar- tensor theories. Any asymmetry in such “gravitational charge” generally leads to the emission of dipolar radiation, which is a particular strong source of energy loss, as it appears at the 1.5 post-Newtonian level in the equations of motion. Of particular interest here are theories that give rise to extra gravitational charges only for BHs and therefore evade any present binary pulsar tests. Certain shift-symmetric Horndeski theories known to have such properties [1412, 1414, 1415, 1428], where a star that collapses to a BH suddenly acquires a scalar charge in a nontrivial manner [1588, 1589] (cf. Sections 2.2 and4.2). Based on mock timing data, Liu et al. [1742] have demonstrated the capability of utilizing a suitable pulsar-BH system to put tight constraints on any form of dipolar radiation.

Finally, there are interesting considerations how a pulsar-BH system could be used to constrain quantum effects related to BHs. For instance, there could be a change in the orbital period caused by the mass loss of an enhanced evaporation of the BH, for instance due to an extra spatial dimension. An absence of any such change in the timing data of the pulsar would lead to constraints on the effective length scale of the extra spatial dimension [1745]. If quantum fluctuations of the spacetime geometry outside a BH do occur on macroscopic scales, the radio signals of a pulsar viewed in an (nearly) edge-on orbit around a BH could be used to look for such metric fluctuations. Such fluctuations are expected to modify the propagation of the radio signals and therefore lead to characteristic residuals in the timing data [1746].

Given the prospects and scientific rewards promised by PSR-BH systems, searches are on-going to discover these elusive objects. Pulsars orbiting stellar-mass BHs are expected to be found in or near the Galactic plane. Since binary evolution requires such system to survive two supernova explosions, this implies a low systemic velocity, placing it close to its original birth place. On-going deep Galactic plane surveys, like

those as part in the High Resolution Universe Survey [1747] or upcoming surveys using the MeerKAT or FAST telescopes, clearly have the potential to uncover such systems. Looking at regions of high stellar density, one can expect even to find millisecond pulsars around BHs due to binary exchange interactions, making globular clusters [1748] and the Galactic Centre [1749] prime targets for current and future surveys. As discussed, finding a pulsar orbiting Sgr A* would be particularly rewarding. Past surveys are likely to have been limited by sensitivity and scattering effects due to the turbulent interstellar medium although the discovery of a radio magnetar in the Galactic Center [1750] indicates that the situation may be more complicated than anticipated [1751]. Searches with sensitive high-frequency telescopes will ultimate provide the answer [1752]. Meanwhile, sensitive timing observations of pulsars in Globular cluster can also probe the proposed existence of Intermediate Mass Black Holes (IMBHs) in the cluster centres by sensing how the pulsars “fall” in the cluster potential. In some clusters, prominent claims [1753] can be safely refuted [1754], while in other clusters IMBHs may still exist in the centre [1755,1756] or their potential mass can at least be constrained meaningfully [1757].

In summary, current and future radio pulsars observations have the potential to study BH properties over a wide mass range, from stellar-mass to super-massive BHs providing important complementary observations presented in this chapter;