Black holes, gravitational waves and fundamental physics: a roadmap

Gravity around BHs is so extreme that gravitational energy is converted into EM and kinetic energy with high efficiency, when gas and/or stars skim the event horizon of astrophysical BHs. Black holes of stellar origin (sBHs) with masses close to those of known stars power galactic x-ray sources in binaries, while supermassive black holes (SMBHs) with masses up to billions of solar masses power luminous quasars and active nuclei at the centre of galaxies. BHs are key sources in EM in our cosmic landscape.

According to General Relativity (GR), Kerr BHs, described by their mass $M_{\rm BH}$ and spin vector ${\bf S}=\chi_{\rm spin} \, G \, M_{\rm BH}/c$ (with $-1\leqslant \chi_{\rm spin}\leqslant 1$) are the unique endstate of unhalted gravitational collapse. Thus understanding astrophysical BHs implies understanding the conditions under which gravitational equilibria lose their stability irreversibly. The chief and only example we know is the case of NSs which can exist up to a maximum mass $M_{\rm max}^{\rm NS}$, around $\newcommand{\msun}{\,{\rm M_\odot}} 2.2\msun$–$\newcommand{\msun}{\,{\rm M_\odot}} 2.6\msun$. No baryonic microphysical state emerges in nuclear matter, described by the standard model, capable to reverse the collapse to a BH state, during the contraction of the iron core of a supernova progenitor. The existence of $M_{\rm max}^{\rm NS}$ is due to the non linearity of gravity which is sourced not only by the mass 'charge' but also by pressure/energy density, according to the Oppenheimer-Volkoff equation. Thus, sBHs carry a mass exceeding $M_{\rm max}^{\rm NS}$. Discovering sBHs lighter than this value (not known yet to high precision) would provide direct evidence of the existence of PBHs arising from phase transitions in the early universe.

As of today, we know formation scenarios in the mass range between $\newcommand{\msun}{\,{\rm M_\odot}} 5\msun$–$\newcommand{\msun}{\,{\rm M_\odot}} 40\msun$, resulting from the core-collapse of very massive stars. The high masses of the sBHs revealed by the LVC, up to $\newcommand{\msun}{\,{\rm M_\odot}} 36\msun$, hint formation sites of low-metallicity, Z¹³⁴, below $0.5\%$ of the solar value $Z_\odot=0.02$ [25–27]. Theory extends this range up to about 40–$\newcommand{\msun}{\,{\rm M_\odot}} 60\msun$ [28] and predicts the existence of a gap, between about $\newcommand{\simless}{\lesssim} \newcommand{\msun}{\,{\rm M_\odot}} 60 \, \simless \, M_{\rm BH}/\msun \, \simless \, 150,$ since in this window pair instabilities during oxygen burning lead either to substantial mass losses or (in higher mass stellar progenitors) the complete disruption of the star [29–31]. sBHs heavier than $\newcommand{\msun}{\,{\rm M_\odot}} 150\msun$ can form at $Z \, < \, 1\% \, Z_\odot$, if the initial mass function of stars extends further out, up to hundreds of solar masses.

The majestic discovery of BBHs, detected by LVC interferometers [1, 2, 5, 11, 18, 20], at the time of their coalescence further indicates, from an astrophysical standpoint, that in nature sBHs have the capability of pairing to form binary systems, contracted to such an extent that GW emission drives their slow inspiral and final merger, on observable cosmic timescales. As GWs carry exquisite information on the individual masses and spins of the BHs, and on the luminosity distance of the source, detecting a population of coalescing sBHs with LVC in their advanced configurations, and with the next-generation of ground-based detectors [32, 33], will let us reconstruct the mass spectrum and evolution of sBHs out to very large redshifts.

Observations teach us that astrophysical BHs interact with their environment, and that there are two ways to increase the mass: either through accretion, or through a merger, or both. These are the two fundamental processes that drive BH mass and spin evolution. Accreting gas or stars onto BHs carry angular momentum, either positive or negative, depending on the orientation of the disk angular momentum relative to the BH spin. As a consequence the spin changes in magnitude and direction [34–36]. In a merger, the spin of the new BH is the sum of the individual and orbital angular momenta of the two BHs, prior to merging [37, 38]. An outstanding and unanswered question is can sequences of multiple accretion-coalescence events let sBHs grow, in some (rare) cases, up to the realm of SMBHs? If this were true, the 'only' collapse to a BH occurring in nature would be driven by the concept of instability of NSs at $M_{\rm max}^{\rm NS}$.

SMBHs are observed as luminous quasars and active galactic nuclei, fed by accretion of gas [39], or as massive dark objects at the centre of quiescent galaxies which perturb the stellar and/or gas dynamics in the nuclear regions [40]. The SMBH mass spectrum currently observed extends from about $\newcommand{\msun}{\,{\rm M_\odot}} 5\times 10^4\msun$ (the SMBH in the galaxy RGG118 [41]) up to about $\newcommand{\msun}{\,{\rm M_\odot}} 1.2\times 10^{10}\msun$ (SDSS J0100+2802 [42]), as illustrated in figure 1. The bulk of active and quiescent SMBHs are nested at the centre of their host galaxies, where the potential well is the deepest. The correlation between the SMBH mass $M_\bullet$ and the stellar velocity dispersion $\sigma$ in nearby spheroids, and even in disk/dwarf galaxies [43] hints towards a concordant evolution which establishes in the centre-most region controlled by powerful AGN outflows. Extrapolated to lower mass disk or dwarf galaxies, this correlation predicts BH masses of $\newcommand{\msun}{\,{\rm M_\odot}} M_\bullet\sim 10^3\msun$ at $\sigma$ as low as $\newcommand{\kms}{\,{\rm km\,s^{-1}}} 10\kms$, typical of nuclear star clusters (globular clusters) [44]. We remark that only BHs of mass in excess of $\newcommand{\msun}{\,{\rm M_\odot}} 10^3\msun$ can grow a stellar cusp. The lighter BHs would random walk, and thus would have a gravitational sphere of influence smaller than the mean stellar separation and of the random walk mean pathlength.

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Observations suggest that SMBHs have grown in mass through repeated episodes of gas accretion and (to a minor extent) through mergers with other BHs. This complex process initiates with the formation of a seed BH of yet unknown origin [45]. The concept of seed has emerged to explain the appearance of a large number of SMBHs of billion suns at $z\sim 6$, shining when the universe was only 1 Gyr old [46]. Furthermore, the comparison between the local SMBH mass density, as inferred from the $M_\bullet-\sigma$ relation, with limits imposed by the cosmic x-ray background light, resulting from unresolved AGN powered by SMBHs in the mass interval between $\newcommand{\msun}{\,{\rm M_\odot}} 10^{8-9}\msun$, indicates that radiatively efficient accretion played a large part in the building of SMBHs below $z\sim 3,$ and that information is lost upon their initial mass spectrum [47]. Thus, SMBHs are believed to emerge from a population of seeds of yet unconstrained initial mass, in a mass range intermediate between those of sBHs and SMBHs, about $\newcommand{\msun}{\,{\rm M_\odot}} 10^2\msun$ to $\newcommand{\msun}{\,{\rm M_\odot}} 10^5\msun$, and therefore they are sometimes dubbed Intermediate-mass BHs (IMBHs).

Seeds are IMBHs that form 'early' in cosmic history (at redshift $z\sim 20$, when the universe was only 180 Myr old). They form in extreme environments, and grow over cosmic time by accretion and mergers. Different formation channels have been proposed for the seeds [45, 48, 49]. Light seeds refer to IMBHs of about $\newcommand{\msun}{\,{\rm M_\odot}} 100\msun$ that form from the relativistic collapse of massive Pop III stars, but the concept extends to higher masses, up to  ∼$\newcommand{\msun}{\,{\rm M_\odot}} 10^3\msun$. These seeds likely arise from runaway collisions of massive stars in dense star clusters of low metallicity [50, 51], or from mergers of sBHs in star clusters subjected to gas-driven evolution [52]. The progenitors of light seeds are massive stars. However, they could be also the end result of repeated mergers among BHs [53]. Finding merging sBHs with the LVC detectors with masses in the pair instability gap would be a clear hint of a second generation of mergers resulting from close dynamical interactions.

Accretion on sBHs occurs in x-ray binaries, and there is no evidence of accretion from the interstellar medium onto isolated sBHs in the Milky Way. But, in gas-rich, dense environments characteristic of galaxy halos at high redshifts, single sBHs might accrete to grow sizably, despite their initial small gravitational sphere of influence, if specific dynamical conditions are met. For instance, in rare cases they may be captured in dense gas clouds within the galaxy [55]. Another possibility is that a sBH forms at the very center of the galaxy, where large inflows may temporarily deepen the potential well and allow it to grow significantly. This 'winning sBH' must be significantly more massive than all other sBHs in the vicinity to avoid being ejected by scatterings and to be retained at the center of the potential well by dynamical friction. Similar conditions can also be present in nuclear star clusters characterized by high escape velocities. After ejection of the bulk of the sBHs, the only (few) remaining isolated BH can grow by tidally disrupting stars and by gas accretion [57] sparking their growth to become an IMBH.

Heavy seeds refer instead to IMBHs of about $\newcommand{\msun}{\,{\rm M_\odot}} 10^{4-5}\msun$ resulting from the monolithic collapse of massive gas clouds, forming in metal-free halos with virial temperatures $\newcommand{\simgreat}{\gtrsim} T_{\rm vir} \, \simgreat \, 10^4$ K, which happen to be exposed to an intense H₂ photodissociating ultraviolet flux [49, 56, 58–60]. These gas clouds do not fragment and condense in a single massive proto-star which is constantly fueled by an influx of gas that lets the proto-star grow large and massive. Then, the star contracts sizably and may form a quasi-star [61], or it may encounter the GR instability that leads the whole star to collapse directly into a BH. Heavy seeds might also form in major gas-rich galaxy mergers over a wider range of redshifts, as mergers trigger massive nuclear inflows [62]. Figure 1 is a cartoon summarising the current knowledge of BHs in our Universe, and the link that may exist between sBHs and SMBHs, which is established by seed BHs along the course of cosmic evolution.

The seeds of the first SMBHs are still elusive to most instruments that exist today, preventing us to set constraints on their nature. Seed BHs are necessarily a transient population of objects and inferring their initial mass function and spin distribution from observations is possible only if they can be detected either through EM or GW observations at very high z, as high as  ∼20 (even $z\sim 40$ as discussed recently). Since, according to GR, BHs of any flavour captured in binaries are loud sources of GWs at the time of their merging, unveiling seeds and MBHs through cosmic ages via their GW emission at coalescence would provide unique and invaluable information on the BH genesis and evolution. The Gravitational Wave Universe is the universe we can sense using GWs as messengers [63, 64]. In this universe, BBHs are key sources carrying invaluable information on their masses, spins and luminosity distance that are encoded in the GW signal. There is one key condition that needs to be fulfilled: that the BHs we aim at detecting pair and form a binary with GW coalescence time smaller that the Hubble time, possibly close to the redshift of their formation. This condition, enabling the detection of seeds at very high redshifts, is extremely challenging to be fulfilled. BHs in binaries form at 'large' separation. Thus, nature has to provide additional dissipative processes leading to the contraction of the BBH down to the scale where GWs drive the inspiral. This requires a strong coupling of the two BHs with the environment, before and after forming a binary system. As we now discuss, understanding this coupling is a current challenge in contemporary astrophysics, cosmology and computational physics [65].

Due to the weakness of gravity, BBH inspirals driven by GW emission occur on a timescale:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nonumber t_{\rm coal} &= {\frac{5 c^5}{256G^3}} {{\mathcal G}}(e)(1-e^2)^{7/2} {\frac{{a}^4}{\nu\, M_{\rm BBH}^3}} \nonumber \\ &= {\frac{5\cdot 2^4}{256}} {{\mathcal G}}(e)(1-e^2)^{7/2}{\frac{GM_{\rm BBH}}{\nu c^3}{\tilde a}^4}, \label{tcoal} \nonumber \end{align} \tag{ 1 }$$

where $M_{\rm BBH}$ is the total mass of the BBH, a and e the semi-major axis and eccentricity respectively (${{\mathcal G}}(e)$ a weak function of e, and ${{\mathcal G}}(0)=1$) and $\nu=\mu/M_{\rm BBH}$ the symmetric mass ratio ($\nu=1/4$ for equal mass binaries), with $\mu$ the reduced mass of the binary. The values of a and e at the time of formation of the binary determine $t_{\rm coal}$, and this is the longest timescale. A (circular) binary hosting two equal-mass seed BHs of $\newcommand{\msun}{\,{\rm M_\odot}} 10^3\msun$ ($\newcommand{\msun}{\,{\rm M_\odot}} M_{\rm BBH}=10^5\msun$) would reach coalescence in 0.27 Gyrs, corresponding to the cosmic time at redshift $z\sim 15,$ if the two BHs are at an initial separation of $a\sim \nu^{1/4} 4.84\times 10^4 R_{\rm G}$ ($\nu^{1/4} \,1.5\times 10^4 R_{\rm G}$) corresponding to $a\sim \nu^{1/4}\,0.1$ AU, ($\nu^{1/4}\,30$ AU). For the case of two equal-mass MBHs of $\newcommand{\msun}{\,{\rm M_\odot}} 10^6\msun$ coalescing at $z\sim 3$ (close to the peak of the star-formation rate and AGN rate of activity in the universe) $a\sim \nu^{1/4} 4.84\times 10^6 R_{\rm G}$ corresponding to about one milli-parsec. These are tiny scales, and to reach these separations the binary needs to harden under a variety of dissipative processes. The quest for efficient mechanisms of binary shrinking, on AU-scales for sBHs and BH seeds, and sub-galactic scales for MBHs, make merger rate predictions extremely challenging, as Nature has to set rather fine-tuned conditions for a BBH to reach these critical separations. Only below these critical distances the binary contracts driven by GW emission. The merger occurs when the GW frequency (to leading order equal to twice the Keplerian frequency) reaches a maximum value,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\msun}{\,{\rm M_\odot}} \displaystyle f_{\rm GW}^{\rm max}\sim {\frac{c^3} {\pi 6^{3/2} GM_{\rm BBH}}}=4.4\times 10^3 \left ({\frac{\msun}{M_{\rm BBH}}}\right)~{\rm Hz}. \label{fmax} \nonumber \end{align} \tag{ 2 }$$

This frequency $f_{\rm GW}^{\rm max}$ scales with the inverse of the total mass of the binary, $M_{\rm BBH}$ as it is determined by the size of the horizon of the two BHs, at the time of coalescence.

Coalescing sBHs in binaries occur in galactic fields [25, 66], or/and in stellar systems such as globular clusters or/and nuclear star clusters [67–71]. Thus, sBHs describe phenomena inside galaxies. Since there is a time delay between formation of the binary and its coalescence, dictated by the efficiency of the hardening processes, sBHs can merge in galaxies of all types, as in this lapse time that can be of the order of Gyrs, host galaxies undergo strong evolution. Instead, coalescing IMBHs, seeds of the SMBHs, formed in DM halos at high redshifts, and thus track a different environment. Forming in pristine gas clouds their pairing either requires in situ formation, e.g. from fissioning of rotating super-massive stars [72], or via halo-halo mergers on cosmological scales subjected to rapid evolution and embedded in a cosmic web that feed baryons through gaseous stream [73]. Coalescing MBHs refer exclusively to galaxy-galaxy mergers of different morphological types [65] occurring during the clustering of cosmic structures, which encompass a wide range of redshifts, from $z\sim 9$ to $z\sim 0$ passing through the era of cosmic reionization, and of cosmic high noon when the averaged star formation rate has its peak.

In the following sections we describe in detail the different channels proposed for the formation and pairing of BBHs at all scales. For each physical scenario we review the state of the art, challenges and unanswered questions and the most promising lines of research for the future. Ample space is devoted to stellar mass objects (BNS and BH-NS binaries and BBHs, with a particular focus on the latter), for which we discuss separately the three main formation channels: pairing of isolated binaries in the field, the various flavors of dynamical formation processes, relics from the early universe. We then move onto discuss the state of the art of our understanding of MBH binary pairing and evolution, the current theoretical and observational challenges, and the role of future surveys and pulsar timing arrays (PTAs) in unveiling the cosmic population of these elusive systems.

The LIGO/Virgo detections of BBH mergers can be explained with stellar-origin BHs [27] or by primordial BHs that have formed from density fluctuations right after Big Bang [74], Stars of different ages and chemical compositions can form BHs and subsequently BBH mergers. In particular, the first metal-free (population III) stars could have produced BBH mergers in the early Universe ($z\approx10$), while the local ($z\approx 0$–2) Universe is most likely dominated by mergers formed by subsequent generation of more metal-rich population II and I stars [75]. The majority of population I/II stars (hereafter: stars) are found in galactic fields (∼$99\%$) where they do not experience frequent or strong dynamical interactions with other stars. In contrast, some small fraction of stars (∼$1\%$) are found in dense structures like globular or nuclear clusters, in which stellar densities are high enough that stars interact dynamically with other stars. Here, we briefly summarize basic concepts of isolated (galactic fields) stellar and binary evolution that leads to the formation of BBH mergers.

4.1.1. Single star evolution.

4.1.2. Binary star evolution.

4.1.3. Reconciling observations and theory.

There seems to be some confusion in the community as to what was expected and predicted by stellar/binary evolution models prior to LIGO/Virgo detections of the first sources. In particular, it is striking that often it is claimed that LIGO/Virgo detections of BBH mergers with very massive BHs were surprising or unexpected. Before 2010, in fact most models were indicating that BNS are dominant GW sources for ground-based detectors (however, see also [132], predicting LIGO detection rates strongly dominated by BBH binaries), and that stellar-origin BHs are formed with small masses of  ∼$\newcommand{\msun}{\,{\rm M_\odot}} 10 \msun$ [146]. The models before 2010 were limited to calculations for stars with high metallicity (typical of the current Milky Way population) and this has introduced a dramatic bias in predictions. However, already around 2010 it was shown that stars at low metallicities can produce much more massive (30–$\newcommand{\msun}{\,{\rm M_\odot}} 80 \msun$) BHs than observed in the Milky Way [96, 147, 148]. Additionally, it was demonstrated that binaries at low metallicities are much more likely to produce BBH mergers than high metallicity stars by one or two ordes of magnitude [25]. This led directly to the pre-detection predictions that (i) the first LIGO/Virgo detection was expected when the detector (BNS) sensitivity range reached about 50–100 Mpc (the first detection was made at 70 Mpc), that (ii) BBH mergers will be the first detected sources, and that (iii) the BBH merger chirp-mass distribution may reach $\newcommand{\msun}{\,{\rm M_\odot}} 30\msun$ [25, 66, 149, 150]. Additionally studies of the future evolution of x-ray binaries like IC10 X-1 and NGC300 X-1 [151] suggested that there exists a large population of merging BH binaries with masses in excess of $\newcommand{\msun}{\,{\rm M_\odot}} 20 \msun$.

Post-detection binary evolution studies expanded on earlier work to show agreement of calculated BBH merger rates and BBH masses with LIGO/Virgo observations [27, 97, 134, 135]. The range of calculated merger rates (10–$\newcommand{\gpy}{{\rm ~Gpc}^{-3} {\rm ~yr}^{-1}} 300\gpy$) comfortably coincides with the observed rate estimates (12–$\newcommand{\gpy}{{\rm ~Gpc}^{-3} {\rm ~yr}^{-1}} 213 \gpy$ for the LIGO/Virgo $90\%$ credible interval). Note that these classical binary evolution rates are typically much higher than rates predicted for dynamical BBH formation channels (5–$\newcommand{\gpy}{{\rm ~Gpc}^{-3} {\rm ~yr}^{-1}} 10 \gpy$, [152, 153]). The most likely detection mass range that is predicted from classical isolated binary evolution is found in the total BBH merger mass range of 20–$\newcommand{\msun}{\,{\rm M_\odot}} 100 \msun$ (e.g. [97]). Examples of merger rate and mass predictions for BBH mergers are given in figures 2 and 3. A similar match between observed LIGO/Virgo BH masses and model predictions is obtained from the dynamical formation channel [152, 153]. Note that this makes these two channels indistinguishable at the moment, although the merger rates are likely to be much smaller for the dynamical channel.

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A caveat of concern for the prospects of LIGO/Virgo detecting BBH mergers with masses above the PISN gap is related to the relatively low GW frequencies of the such massive BBH binaries with chirp masses above $\newcommand{\msun}{\,{\rm M_\odot}} 100\;\msun$. During the in-spiral, the emitted frequencies are expected to peak approximately at the innermost stable circular orbit (ISCO), before the plunge-in phase and the actual merging. Hence, the emitted frequencies are most likely less than 100 Hz, and with redshift corrections the frequencies to be detected are easily lower by a factor of two or more. A frequency this low is close to the (seismic noise) edge of the detection window of LIGO/Virgo and may not be resolved.

The current LIGO/Virgo broad range of an empirically determined local BBH merger-rate density (12–$\newcommand{\gpy}{{\rm ~Gpc}^{-3} {\rm ~yr}^{-1}} 213 \gpy$) can easily be explained by uncertainties in key input physics parameters of population synthesis modelling, such as the slope of the IMF or the efficiency of the CE ejection, see e.g. table 5 in [97]. Alternatively, it may be explained by altering BH natal kicks [27] from full NS natal kicks corresponding to a low rate estimate) to almost no BH kicks (high rate estimate). Once LIGO/Virgo narrows its empirical estimate it may be possible to use the merge-rate density to constrain the input physics applied in modelling, although it should be cautioned that there is a large degree of degeneracy [97, 132].

LIGO/Virgo provides an estimate of the effective spin parameter that measures the projected BH spin components ($a_1, a_2$) parallel to binary angular momentum, weighted by BH masses ($M_1, M_2$):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \chi_{\rm ef\,\!f}\equiv \frac{M_1 a_1 \cos\Theta_1 + M_2 a_2 \cos\Theta_2}{M_1+M_2}, \label{eq.xeff} \nonumber \end{align} \tag{ 3 }$$

where $\Theta_{1,2}$ are the angles between the BH spins and the orbital angular momentum vector. So far, for the six LIGO/Virgo BBH detections the effective spins cluster around $|\chi_{\rm ef\,\!f}| < 0.35$ (e.g. [11]). This defies the expectations for the main BBH formation channels. If spin magnitudes of stellar-origin BHs are significant, as estimated for several galactic and extra-galactic x-ray binaries [154], then the dynamical formation channel (random BH capture) predicts an isotropic $\chi_{\rm ef\,\!f}$ distribution, while the classical binary evolution channel mostly predicts aligned BH spins (aligned stellar spins that are only moderately misaligned by BH natal kicks). Hence, in the latter case one expects a distribution peaked at high values of $\chi_{\rm ef\,\!f}$. On the one hand this tension is rather unfortunate, as it does not allow to distinguish between these two very different scenarios of BBH merger formation. On the other hand, this is a great opportunity to learn something new about stars and BHs that was so far not expected and is not easily understood in the framework of current knowledge.

There are five potential explanations of this apparent tension. First, there could be a mechanism that puts both BH spins in the plane of the binary orbit, producing $\chi_{\rm ef\,\!f}=0$ independent of BH masses and spin magnitudes. Such a mechanism is proposed to operate in triple stars [155]. Note that triple stars are a minority of stars (10–$20\%$ of all field stars) and that the proposed mechanism requires very specific tuning to operate, so it is not clear how likely it is that it worked for all LIGO/Virgo sources. Second, there could be a mechanism that forces the BH spins to be in opposite directions so that they cancel out. For approximately equal mass BBH binaries (typical of LIGO/Virgo sources) this would imply 180 degree flip of spins. No mechanism is known to produce such configurations. Third, both BH spin magnitudes may be very small reducing effective spin parameter to $\chi_{\rm ef\,\!f}=0$, independent of other merger parameters. This was already proposed and is used in studies of angular momentum transport in stars [122]. Fourth, LIGO/Virgo BHs may not have been produced by stars, but for example they come from a primordial population for which small spins are naturally expected [74, 156]. Fifth, it may be the case that the spin of a BH (at least its direction) is not mainly determined by the angular momentum of the progenitor star, but a result of the physics of the collapse (spin tossing, e.g. [157]). In that case, there is no reason to assume the spins in mergers formed from isolated binary evolution are aligned and they may be isotropic. Note that with these five options, that need to be tested and developed further, one cannot determine the main formation channel of BBH mergers, as it was proposed in several recent studies [145, 158–160].

The issue of spins is rather fundamental as the effective spin parameter most likely contains information on natal BH spin magnitudes and therefore information on stellar astrophysics regarding angular momentum transport in massive stars, which is still unconstrained by electromagnetic observations. Possibly the second-formed BH spin could have been increased in binary evolution by accretion of mass from a companion star. However, it is argued that BHs cannot accrete a significant amount of mass in the binary evolution leading to the BBH formation [27, 122, 161]. This is partly due to very low accretion rates during a CE of 1–$10\%$ of the Bondi-Hoyle accretion rate [162–165]) and fast timescale Roche-lobe overflow (RLO) in massive progenitor binaries that leads to ejection of most of the exchanged matter from the binary system (due to super-Eddington mass-transfer rates). The amount of mass accreted by BHs in binary systems ($\lesssim$1–$\newcommand{\msun}{\,{\rm M_\odot}} 3 \msun$) cannot significantly spin up massive BHs (10–$\newcommand{\msun}{\,{\rm M_\odot}} 30\msun$) that are detected by LIGO/Virgo.

It is important to note the most challenging parts of the evolutionary predictions in the context of BBH formation. In the classical binary evolutionary channel, the two most uncertain aspects of input physics are related to the CE evolution and the natal BH kicks. Although some observational constraints on both processes exist, they are rather weak. Systems entering CE evolution have recently been reported. However, they are not as massive as stars that could produce NSs or BHs [166]. The search for CE traces as IR outburts has so far yielded no clear detection of emerging or receding x-ray binaries as expected in this scenario [167].

BH natal kicks are only measured indirectly by positions and motions of x-ray binaries hosting BHs, and usually only lower limits on natal kicks are derived [114–116, 168]. On theoretical grounds, reliable models for CE [141] and supernovae [169] are missing. In the chemically homogeneous evolution channel the largest uncertainty is connected with the efficiency of the mixing, the number of massive binaries that can form in very close orbits, and the strength of tidal interactions in close binaries. Since initial orbital period distributions are measured only in the very local Universe [124], it is not clear whether they apply to the majority of all stars and thus it is not fully understood how many stars are potentially subject to this kind of evolution. Even a deeper problem exists with our understanding of tides and their effectiveness in close binaries [144, 145], and effective tides are the main component of input physics in chemically homogeneous evolution.

Astrophysical inferences from GW observations are currently limited. First, it is not known which formation channel (or what mixture of them) produces the known LIGO/Virgo BBH mergers. Since each channel is connected to specific set of conclusions (for example, the isolated binary channel informs about CE evolution and natal kicks; while the dynamical channel informs predominantly about stellar interactions in dense clusters) it is not clear which physics we are testing with GW observations. Second, within each channel there is degeneracy such that multiple model parameters are only weakly constrained by observations. As nobody so far was able to deliver a comprehensive study of the large multi-dimensional parameter space, the inferences on various model parameters (e.g. the strength of BH natal kicks or the CE efficiency) are hindered by various and untested model parameter degeneracies. However, it is already possible to test several aspects of stellar evolution as some processes leave unambiguous signatures in GW observations. For example, the existence of the first and the second mass gap, if confirmed by LIGO/Virgo, will constrain core-collapse supernovae and PISNe, respectively. Careful studies with detailed exposure of caveats are needed to transform future observations into astrophysical inferences.

It is expected that GW events resulting from the merger of stellar-mass BHs are unlikely to produce electromagnetic counterparts. Nevertheless, a (marginal) transient signal detected by the Fermi gamma-ray burst monitor, 0.4 s after GW150914, was reported [170]. This claim encouraged several theoretical speculations for a possible origin. It has been suggested [171] that a tiny fraction of a solar mass of baryonic material could be retained in a circumbinary disk around the progenitor binary which probably shed a total mass of  >$\newcommand{\msun}{\,{\rm M_\odot}} 10\msun$ during its prior evolution. The sudden mass loss and recoil of the merged BH may then shock and heat this material, causing a transient electromagnetic signal. It will be interesting see if any further electromagnetic signals will be associated with BBH mergers in the near future.

The formation of double NSs has been studied in detail following the discovery of the Hulse-Taylor pulsar [172] and currently more than 20 such BNS systems are known—see [157] for a details and a review on their formation and evolution. Whereas no BBH binaries had been detected prior to GW150914, detailed knowledge on BNS systems was known for many years from galactic radio pulsar observations [173].

LIGO/Virgo has currently only detected one BNS merger event (GW170817), located in the lenticular (S0) galaxy NGC 4993, and thus the local empirical BNS merger rate density still remains rather uncertain: $\newcommand{\gpy}{{\rm ~Gpc}^{-3} {\rm ~yr}^{-1}} 1540^{+3200}_{-1220} \gpy$ ($90\%$ credible limits [21]). The study of double NSs is relevant for the study of BHs because it gives independent constraints on the evolution of similar massive binary populations from which binary BHs are formed. In particular the question which stars for NSs and which form BHs and if and how this depends on previous binary interactions is a question that likely only can be answered observationally by significant statistics on the relative abundance of double BHs, BNS and NS-BH binaries.

There are two major sites to produce BNS mergers: isolated binaries in galactic fields (the main contributor), and dense environments in globular and nuclear clusters. None of these sites (nor any combination of them) can easily reproduce the preliminary estimated LIGO/Virgo event rate, even if all elliptical host galaxies are included within the current LIGO/Virgo horizon [174]. The local supernova rate can be estimated to be about $\newcommand{\gpy}{{\rm ~Gpc}^{-3} {\rm ~yr}^{-1}} 10^5 \gpy$, so the current empirical BNS merger rate from LIGO/Virgo would imply a very high efficiency of BNS binary formation.

This apparent tension may be solved if BNS mergers are allowed to originate from a wide range of host galaxies and if the low-end of the LIGO/Virgo merger-rate estimate is used ($\newcommand{\gpy}{{\rm ~Gpc}^{-3} {\rm ~yr}^{-1}} 320 \gpy$). Population synthesis studies seem to agree that rates as high as 200–$\newcommand{\gpy}{{\rm ~Gpc}^{-3} {\rm ~yr}^{-1}} 600 \gpy$ can possibly be reached if favorable conditions are assumed for classical binary evolution [97, 175, 176]. In the coming years, the statistics of the empirical BNS merger rate will improve significantly and reveal whether current theoretical BNS merger rates need a revision. It is interesting to notice, however, that calibrations with the rates of observed short gamma-ray bursts and the rate of mergers required to reproduce the abundances of heavy r-process elements favor a merger-rate density significantly smaller than the current empirical rate announced by LIGO/Virgo [97].

The main uncertainties of the theoretically predicted merger rate of BNS binaries are also related to CE evolution and SNe (similar to the case of BBH mergers). A CE evolution is needed to efficiently remove orbital angular momentum to tighten the binary orbit and allow a merger event within a Hubble time. However, the onset criterion and the efficiency of the in-spiral in a CE remain uncertain [141, 177]. The kick velocities imparted on newborn NSs span a wide range from a few ${\rm km~s}^{-1}$ (almost symmetric SNe) to more than 1000 ${\rm km~s}^{-1}$ and are sometimes difficult to determine [178]. The kick magnitude seems to be related to the mass of the collapsing core, its density structure and the amount of surrounding envelope material [110]. Additional important factors for the predicted merger rates include the slope of the initial-mass function and the efficiency of mass accretion during RLO [97].

All taken together, the predicted merger rate of double NSs in a Milky Way equivalent galaxy vary by more than two orders of magnitude [146]. The empirical merger rate that LIGO/Virgo will detect at design sensitivity in a few years is of uttermost importance for constraining the input physics behind the rate predictions. Besides the detection rates, the mass spectrum and the spin rates of the double NS mergers will reveal important information about their origin. Although their precise values cannot be determined due to degeneracy, the overall distribution of estimated NS masses will reveal information on their formation process (electron capture versus iron-core collapse SNe), as well as constraining the nuclear-matter equation-of-state. The latter will also be constrained from tidal deformations of the NSs in their last few orbits [21].

An important observational signature of the merger event of BNS binaries is the detection of the ring-down signal of either a meta-stable highly massive NS or a BH remnant. Such information would set constraints on the fundamental equation-of-state of nuclear matter at high densities. Whereas LIGO/Virgo is not sensitive enough to detect a ring-down signal, it is the hope that third-generation GW detectors might be able to do so. Another important observational input is the distribution of mass ratios in BNS merger events. This distribution could provide important information about the formation of NSs and the nature of the supernovae (e.g. electron-capture versus iron core-collapse supernovae).

Optical follow-up will in many cases reveal the location of a double NS merger (e.g. [22]). This will provide information on their formation environments [179, 180] and kinematics [181], besides crucial information on heavy r-process nucleosynthesis [182].

The formation of mixed BH/NS mergers is expected to follow similar scenarios as double NS or double BH [129, 132] with all the associated uncertainties.

It is perhaps somewhat surprising that LIGO/Virgo detected a double NS merger (GW170817) before a mixed BH/NS merger, since (at least some) population synthesis codes predict a detection rate of mixed BH/NS systems which is an order of magnitude larger that the expected detection rate of double NS systems (with large uncertainties, e.g. [97]). Hence, if these predictions are correct, GW170817 is a statistical rare event and detections of mixed BH/NS systems are expected already in the upcoming O3/O4 LIGO runs. The detection of mixed BH-NS mergers is interesting for two reasons: (i) a key question is whether BH-NS and NS-BH binaries may be distinguished from one another (i.e. the formation order of the two compact objects, which leads to a (mildly) recycled pulsar in the latter case), and (ii) the detected in-spiral of mixed BH-NS mergers may reveal interesting deviations from GR and pure quadrupole radiation given the difference in compactness between BHs and NSs [183].

The recent GW observations from six BBH mergers (GW150914, LVT151012, GW151226, GW170104, GW170608, GW170814) and a BNS merger (GW170817) opened ways to test the astrophysical theories explaining the origin of these sources [1, 2, 11, 18, 20, 21]. As discussed earlier, the component masses of these merging sources span a range between 8–$\newcommand{\e}{{\rm e}} \newcommand{\Msun}{ \mathrm{M}_{\odot}} 35\Msun$ [11], which is different from the distribution of BHs seen in x-ray binaries, 5–$\newcommand{\e}{{\rm e}} \newcommand{\Msun}{ \mathrm{M}_{\odot}} 17\Msun$ [184] with two possible exceptions (NGC300X-1 and IC10X-1). The event rates of BBH mergers are estimated to be between 40–$\newcommand{\e}{{\rm e}} \newcommand{\Gpc}{{\rm Gpc}} 213~\Gpc^{-3}~{\rm yr}^{-1}$ for a power-law BH mass function and between 12–$\newcommand{\e}{{\rm e}} \newcommand{\Gpc}{{\rm Gpc}} 65~\Gpc^{-3}~{\rm yr}^{-1}$ for a uniform-in-log BH mass function [11], which is higher than previous theoretical expectations of dynamically formed mergers, for instance see [146]. The event rates of BNS mergers is currently based on a single measurement which suggests a very high value of $\newcommand{\e}{{\rm e}} \newcommand{\Gpc}{{\rm Gpc}} 1540^{+3200}_{-1220}\,\Gpc^{-3}~{\rm yr}^{-1}$ [21] (c.f. [174]). How do we explain the observed event rates and the distribution of masses, mass ratios, and spins?

Several astrophysical merger channels have been proposed to explain observations. Here we review some of the recent findings related to dynamics, their limitations and directions for future development. These ideas represent alternatives to the classical binary evolution picture, in which the stars undergo poorly understood processes, such as common envelope evolution. In all of these models the separation between the compact objects is reduced dynamically to less than an AU, so that GWs may drive the objects to merge within a Hubble time, $t_{\rm Hubble}=10^{10}\,~{\rm yr}$.

5.2.1. Dynamical formation and mergers in globular clusters.

Although about 0.25% of the stellar mass is currently locked in globular clusters (GCs) [185–187], dynamical encounters greatly catalyze the probability of mergers compared to that in the field. Within the first few million years of GC evolution, BHs become the most massive objects. Due to dynamical friction, they will efficiently segregate to the cluster center [188] where they can dynamically interact and form binaries with other BHs [189, 190]. The dense environments of GCs can also lead to binary-single and binary-binary encounters involving BHs that could result in their merger. Collisional systems like GCs can also undergo core collapse, during which central densities can become very large leading to many strong dynamical interactions. The encounter rate density is proportional to $\mathcal{R}\sim \int {\rm d}V \,\langle n_{*}^2 \rangle \, \sigma_{\rm cs} v$, where n_* is the stellar number density, $\sigma_{\rm cs}\sim GMb/v^2$ is the capture cross section, M is the total mass, b is the impact parameter, $v$ is the typical velocity dispersion. Note the scaling with $\langle n_*^2\rangle$, where $\newcommand{\e}{{\rm e}} \newcommand{\pc}{{\rm pc}} \langle n_{*}^2\rangle^{1/2} \sim 10^5\pc^{-3}$ in GCs and  ∼$\newcommand{\e}{{\rm e}} \newcommand{\pc}{{\rm pc}} 1\pc^{-3}$ in the field.

Estimates using Monte Carlo method to simulate realistic GCs yield merger rates of at least $\newcommand{\e}{{\rm e}} \newcommand{\Gpc}{{\rm Gpc}} \mathcal{R}_{\rm GC}\sim 5 ~\Gpc^{-3}~{\rm yr}^{-1}$ [152, 153], falling below the current limits on the observed rates. Rate estimates from results of direct N–body simulations also yield a similar value of $\newcommand{\e}{{\rm e}} \newcommand{\Gpc}{{\rm Gpc}} \mathcal{R}_{\rm GC}\sim 6.5 ~ \Gpc^{-3}~{\rm yr}^{-1}$ [191]. In particular, these papers have shown that the low-mass GCs below $10^5\mathrm{M}_{\odot}$ have a negligible contribution to the rates. However, they also show that initially more massive GCs (more massive than $10^6\mathrm{M}_{\odot}$) contribute significantly to the rates. [153] argue that actual merger rates from BHs originating in GCs could be 3–5 times larger than their estimated value of  ∼$\newcommand{\e}{{\rm e}} \newcommand{\Gpc}{{\rm Gpc}} 5 ~\Gpc^{-3}~{\rm yr}^{-1}$ due to uncertainties in initial GC mass function, initial mass function of stars in GCs, maximum initial stellar mass and evolution of BH progenitors. Furthermore, BBH merger rates can be significant in young clusters with masses  ∼$\newcommand{\e}{{\rm e}} \newcommand{\Msun}{ \mathrm{M}_{\odot}} 10^4\,\Msun$ [50, 192].

A simple robust upper limit may be derived by assuming that all BHs merge once in each GC in a Hubble time:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Mpc}{{\rm Mpc}} \newcommand{\Gpc}{{\rm Gpc}} \newcommand{\Msun}{ \mathrm{M}_{\odot}} \displaystyle \mathcal{R} & \leqslant \frac12f_{\rm BH} N_{*}\frac{n_{\rm GC}}{t_{\rm Hubble}} \nonumber \\ & < \frac12 \frac{\int_{20\Msun}^{150\Msun} f_{\rm IMF}(m) {\rm d}m}{\int_{0.08\Msun}^{150\Msun} m f_{\rm IMF}(m) {\rm d}m} \times 10^{5.5}\Msun \times\frac{0.8\Mpc^{-3}}{10^{10}~{\rm yr}} \nonumber \\ &= 38\,\Gpc^{-3}~{\rm yr}^{-1} \label{eq:GCrate} \nonumber \end{align} \tag{ 4 }$$

where $f_{\rm BH}$ is the fraction of stars that turn into BHs from a given stellar initial mass function $f_{\rm IMF}$, N_* is the initial number of stars in a GC, and $n_{\rm GC}$ is the cosmic number density of GCs, and $\newcommand{\e}{{\rm e}} \newcommand{\Msun}{ \mathrm{M}_{\odot}} f_{\rm IMF}(m) \propto (m/0.5\Msun){}^{-2.3}$ for $\newcommand{\e}{{\rm e}} \newcommand{\Msun}{ \mathrm{M}_{\odot}} m>0.5\Msun$ and $\newcommand{\e}{{\rm e}} \newcommand{\Msun}{ \mathrm{M}_{\odot}} (m/0.5\Msun){}^{-1.3}$ otherwise [193]. The result is not sensitive to the assumed upper bound on mass of the BH progenitor, which is set by the pair instability supernova. Recent estimates find 110–$\newcommand{\e}{{\rm e}} \newcommand{\Msun}{ \mathrm{M}_{\odot}} 130\,\Msun$ for GC metallicities [31]. However, note that the mass in GCs may have been much higher than currently by a factor  ∼5, since many GCs evaporated or got tidally disrupted [194, 195]. This effect increases the rates by at most a factor two at z  <  0.3, but more than a factor 10 at z  >  2.5 [196].

5.2.2. Dynamical and relativistic mergers in galactic nuclei.

5.2.3. Mergers facilitated by ambient gas.

5.2.4. Isolated triples.

5.2.5. Mergers in dark matter halos.

Recent years have brought significant advances in modelling the dynamical environments leading to GW events.

5.3.1. Direct N-body integration.

5.3.2. Monte-Carlo methods.

5.3.3. Secular symplectic N-body integration.

5.3.4. Semianalytical methods.

How can dynamical models test the astrophysical interpretation of GW sources? GW detectors can measure the binary component masses, the spin magnitudes and direction, the binary orbital plane orientation, the eccentricity, the distance to the source and the sky location. The observed distributions of these parameters may be compared statistically to the predicted distributions for each channel. Some smoking gun signatures of the astrophysical environment are also known for individual sources, discussed below.

5.4.1. Mass distribution.

The mass distribution of mergers depends on the theoretically poorly known BH mass function times the mass-dependent likelihood of mergers. Particularly, the mergers of massive objects are favored in GCs, due to the mass dependence of binary formation in triple encounters, binary exchange interactions, dynamical friction, and the Spitzer instability. Monte Carlo and N-body simulations show that the likelihood of merger is proportional to M⁴ in GCs [254]. The 2-dimensional total mass and mass ratio distribution of mergers may be used to test this prediction [255] and to determine the underlying BH mass function in these environments. The mass function of mergers may also vary with redshift as the most massive BHs are ejected earlier [152].

Recently, [256] introduced a parameter to discriminate among different astrophysical channels:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \alpha = -(m_1+m_2)^2 \frac{\partial^2 \mathcal{R}}{\partial m_1\partial m_2}. \nonumber \end{align} \tag{ 5 }$$

This dimensionless number is $1 \pm 0.03$ for PBHs formed dynamically in the early universe (if the PBH mass function is assumed to be flat. For arbitrary PBH mass function the results can be substantially different [257]), 10/7 for BHs which form by GW emission in collisionless systems such as DM halos. For BHs which form by GW emission in collisional systems which exhibit mass segregation, this $\alpha$ varies for different component masses. In galactic nuclei it ranges between 10/7 for the low mass components in the population to  −5 for the highest mass component. It would be very useful to make predictions for $\alpha$ for all other merger channels.

5.4.2. Spin distribution.

5.4.3. Eccentricity distribution.

5.4.4. Sky location distribution.

5.4.5. Smoking gun signatures.

The nature of dark matter (DM) is one of the most pressing open problems of modern cosmology. The evidence for this mysterious form of matter started to grow in the early 20th century [278], and it is today firmly established thanks to a wide array of independent observations [279]. Its nature is still unknown, and the candidates that have been proposed to explain its nature span over 90 orders of magnitude in mass, from ultra-light bosons, to massive BHs [280–282]. An intriguing possibility is that at least a fraction of DM is in the form of PBHs [283]. The recent LVC detections [225, 284], have revived the interest in these objects, and prompted a reanalysis of the existing bounds [283, 285] and new prospects for phenomenological signatures [156, 286].

PBHs might be produced from early universe phase transitions, topological defects (e.g. cosmic strings, domain walls), condensate fragmentation, bubble nucleation and large amplitude small scale fluctuations produced during inflation (see the reviews [229, 287] and references therein). Excluding the possibility of BH relics, non-evaporating PBHs could range in mass from 10⁻¹⁸ to 10⁶ solar masses today, although the most interesting range today is that associated with the observed LIGO BBHs, around a few to tens of solar masses. For those PBHs, amongst the most promising production mechanisms was the one first proposed in [288] from high peaks in the matter power spectrum generated during inflation. When those fluctuations re-enter the horizon during the radiation era, their gradients induce a gravitational collapse that cannot be overcome even by the radiation pressure of the expanding plasma, producing BHs with a mass of order the horizon mass [289]. Most of these PBH survive until today, and may dominate the Universe expansion at matter-radiation equality.

One important question is what fraction $f_{\rm PBH}= \Omega_{\rm PBH}/\Omega_{\rm DM}$ of DM should be made of O(10M) PBHs in order to match the BBH merger rate inferred by LIGO/Virgo ($\mathcal{R} \simeq 10$–200 Gpc⁻³ yr⁻¹ [11]). If one considers PBH binaries that form within virialized structures, the corresponding rate is compatible with $f_{\rm PBH} = 1$ [225]. However, PBH binary systems can form in the early Universe (deep in the radiation era) as well [290, 291], if the orbital parameters of the pair allow the gravitational pull to overcome the Hubble flow and decouple from it. A recent calculation of the associated merger rate today [228]—significantly extended in [227]—provides a much larger estimate (compared to the former scenario): This result can be translated into a bound on $f_{\rm PBH}$, which is potentially stronger than any other astrophysical and cosmological constraint in the same range. The constraint has been recently put on more solid grounds in [292] by taking into account the impact of the DM mini-halos around PBHs, which have a dramatic effect on the orbits of PBH binaries, but surprisingly subtle effect on their merger rate today. Several aspects of this calculation are still under debate, including the PBH mass distribution, the role of a circumbinary accretion disk of baryons, the impact of initial clustering of PBHs (see [293]) and the survival until present time of the binary systems that decoupled in the radiation era.

The critical collapse threshold to form a PBH is typically taken to be $\newcommand{\e}{{\rm e}} \delta_c\equiv {\delta\rho_c/\rho}\sim0.5$, with an exponential sensitivity on the background equation of state, the initial density profile and angular momentum. Because any initial angular momentum suppresses PBH formation, PBHs are expected to spin slowly, a potential way to discriminate between them and (rapidly rotating) astrophyical BHs [294].

Inflationary models which generate PBHs are required to generate a much larger amplitude of perturbations on small scales compared to those observed on CMB scales. This can be achieved either through multiple-field effects or an inflection point in single-field inflation [295, 296]. These typically generate a reasonably broad mass fraction of PBHs, in contrast to the monochromatic mass spectrum usually assumed when interpreting observational constraints. In addition, the softening of the equation of state during the QCD phase transition greatly enhances the formation probability of solar mass PBHs compared to the more massive merging BHs LIGO detected [297]. BHs below the Chandrasekhar mass limit would be a smoking gun of a primordial origin.

At non-linear order, scalar and tensor perturbations couple, implying that the large amplitude perturbations required to generate PBHs will also generate a stochastic background of GWs. For the LIGO mass range the corresponding GW frequency is constrained by pulsar timing arrays unless the scalar perturbations are non-Gaussian [298, 299].

Constraints on the PBH abundance are typically phrased in terms of $f_{\rm PBH} = \Omega_{\rm PBH}/\Omega_{\rm DM}$, the fraction of the total DM density which can be contributed by PBHs. Even in the case where $f_{\rm PBH} < 1$, a future detection via astrophysical probes or GW searches remains hopeful [228]. We outline selected astrophysical constraints below, summarizing these in figure 4, where we focus on PBHs around the Solar mass range, most relevant for GW signals.

Zoom In Zoom Out Reset image size

Micro-lensing: The MACHO [300] and EROS [301] collaborations searched for micro-lensing events in the Magellanic Clouds in order to constrain the presence of Massive Compact Halo Objects (MACHOs) in the Milky Way Halo. Considering events on timescales of $\mathcal{O}$(1–800) d constrains $f_{\rm PBH} < 1$ for masses in the range $M_{\rm PBH} \in [10^{-7}, 30]\,M_\odot$ (though these constraints come with a number of caveats, see e.g. [285, 302]). Another promising target is M31: an important fraction of the Andromeda and Milky Way DM halo can be probed by micro-lensing surveys, and an interesting hint comes from the observation of 56 events in the $\mathcal{O}$(1–100) d range from this region of interest, which may suggest a MACHO population with $f \sim 0.5$ and mass $1 \,M_\odot$ or lighter [303, 304]. A recent search for lensing of Type Ia Supernovae obtained constraints on all PBH masses larger than around $10^{-2}\,M_\odot$, although substantial PBH fractions $f_{\rm PBH} \lesssim 0.6$ are still compatible with the data [305]. For wide mass distributions the SN lensing constraints go away [74] and PBH could still constitute all of the DM.

Early Universe constraints: PBHs are expected to accrete gas in the Early Universe, emitting radiation and injecting energy into the primordial plasma. This in turn can lead to observable distortions of the CMB spectrum and can affect CMB anisotropies [306]. Early calculations over-estimated the effect [307, 308], with more recent studies finding weaker constraints [309, 310]. In spite of this, data from COBE/FIRAS [309, 311] and PLANCK [310] can still rule out PBHs with masses $M_{\rm PBH} \gtrsim 100 \,M_\odot$ as the dominant DM component. It should be noted that these constraints depend sensitively on the details of accretion onto PBHs in the Early Universe (see e.g. [312]).

Dynamical constraints: The presence of PBHs is also expected to disrupt the dynamics of stars. Wide halo binaries may be perturbed by PBHs and the actual observation of such systems constrains the PBHs abundance above $\simeq$$20 \,M_\odot$ [313] (although significant fractions $f_{\rm PBH} \lesssim 0.2$ are still allowed). PBHs are also expected to dynamically heat and thereby deplete stars in the centre of dwarf galaxies. Observations of stellar clusters in Eridanus II [226] and Segue I [314] have been used to constrain PBHs heavier than $\mathcal{O}(1)\,M_\odot$ to be sub-dominant DM components, unless they have an IMBH at their center [315].

Radio and x-ray constraints: If PBHs exist in the inner part of the Galaxy, which contains high gas densities, a significant fraction of them would inevitably form an accretion disk and emit a broad-band spectrum of radiation. A comparison with existing catalogs of radio and x-ray sources in the galactic ridge region already rules out the possibility that PBHs constitute all of the DM in the Galaxy, even under conservative assumptions on the physics of accretion [316]. During the next decade, the SKA experiment will provide an unprecedented increase in sensitivity in the radio band; in particular, SKA1-MID will have the unique opportunity to probe the presence of a subdominant population of PBHs in the Galaxy in the $10 \div 100 \, M_\odot$ mass range, even if it amounts to a fraction as low as $\simeq$1% of the DM.

In figure 4, we show only the most conservative of these constraints, which suggest that PBHs may still constitute all of the DM for masses close to $10\,M_\odot$, or a smaller fraction ($f_{\rm PBH} \gtrsim 0.1$) for masses up to $\mathcal{O}(100\,M_\odot)$. We highlight that such astrophysical constraints may have large systematic uncertainties and that these constraints apply only to PBHs with a mono-chromatic mass function. Recent studies have begun re-evaluating these constraints for extended mass functions [285, 317–319]. For physically motivated mass functions, it may be possible to achieve up to $f_{\rm PBH} \sim 0.1$ for PBHs in the mass range 25–$100 \,M_\odot$ [320]. Relaxing certain dynamical constraints (which typically have large uncertainties) or considering more general mass functions may accommodate an even larger PBH fraction [320, 321].

A number of observations may help discriminating PBHs from ordinary astrophysical BHs. The detection of GWs from a binary system including a compact object lighter than standard stellar BHs, say below one solar mass, would point towards the existence of PBHs. This can be deduced from the highest frequency reached in the GW chirp signal, $f_{\rm ISCO} = 4400~{\rm Hz}\, (M_\odot/M)$, and it is in principle possible already with Advanced LVC in the next run O3 [286]. The detection of GWs at redshift $z \gtrsim 40$ would imply a non-standard cosmology, or the existence of PBHs [322]. Further insights on the origin of BHs might be obtained through the analysis of 'environmental' effects, which are discussed in section 5, and through the analysis of the spatial distribution and mass function of x-ray and radio sources powered by BHs.

The cosmic merger history of SMBBHs is expected to form a low-frequency GWB, which may be detected by PTAs in the next few years [391–393]. The time to detection depends strongly on the number of pulsars in the array, the total length of the dataset, and the underlying astrophysics which affects the SMBBH mergers, such as stellar hardening process, interactions with accretion disks, binary eccentricity, and potentially SMBH stalling. Searches for the GWB have resulted in evermore stringent limits on the amplitude A of the GWB reported at a reference frequency of 1 yr⁻¹:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle h_c = A \left( \frac{f}{\mathrm{yr^{-1}}}\right)^{-2/3} \, , \nonumber \end{align} \tag{ 6 }$$

where h_c is the characteristic strain of the GWB [394]. This simple power-law scaling assumes the the SMBBH systems are circular when emitting GWs, and that they are fully decoupled from their environment. Binary eccentricity and interactions with gas and stars around the binary can deplete the GW signal at very low frequencies (equivalently at wide binary separations), causing the GW strain spectrum to turn over [395–398]. On the other hand, the amplitude of the GWB is affected by the abundance and mass range of the cosmic population of SMBHs. Therefore, future detections of a stochastic GWB will allow to constrain both the overall population of SMBBHs and the physics driving their local dynamics [399–405]. Current non-detections have been used in [406] to challenge the popular $M_{\rm BH}-M_{\rm bulge}$ from [40]. However, a full analysis taking into account uncertainties in the merger rate, SMBBH dynamics and the possibility of stalling is needed to draw firm conclusions [407].

The current upper limit on A from the various PTA experiments are similar and are improving with time. From the EPTA is $A<3\times 10^{-15}$ [408], from the PPTA this is $A<1\times 10^{-15}$ [389], from NANOGrav this is $A <1.45\times 10^{-15}$ [409], and the IPTA limit is $1.7\times 10^{-15}$ [390]. In order to improve those, further more sensitive observations are needed, which can and will be provided by improved instrumentation and new telescopes like FAST, MeerKAT and eventually the SKA [410]. Additionally, systematics needs to be addressed. For instance, solar system ephemeris errors can mimic a GWB signal, if the underlying data are sufficiently sensitive [411]. Here, mitigation techniques can be applied, as already done in the recent analysis of the NANOGrav 11 year data [409], while other effects, such as the interstellar weather may be best addressed with multi-frequency observations. Future IPTA results will take those and other effects into account.

Individual nearby and very massive SMBBHs emitting nHz GWs can also be detected with PTA experiments [414–417]. These SMBBHs are likely in giant elliptical galaxies, though the timescale between the galaxy merger and the subsequent SMBH merger is poorly understood. Unlike LIGO or LISA sources, these SMBBHs are in the PTA band for many millions of years, and will likely merge outside both the PTA and LISA band. The sky and polarization averaged strain of a continuous GW source is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle h = \sqrt{\frac{32}{5}}\frac{\mathcal{M}_c^{5/3}[\pi f (1+z)]^{2/3}}{D_L} \, , \nonumber \end{align} \tag{ 7 }$$

where $\mathcal{M}_c^{5/3} = [q/(1+q){}^2] M^{5/3}$ is the binary chirp mass, q  <  1 is the binary mass ratio, M is the total binary mass, f  is the GW frequency and D_L is the luminosity distance to the binary.

The time to detection of continuous GW sources has been estimated in various ways: namely by simulating a GW source population based on cosmological simulations [371, 392, 418], or by using an underlying galaxy catalog to estimate which nearby galaxies can host SMBBH systems which are PTA targets [372, 419]. Both these approaches largely agree that at least one SMBBH system will be detected in the next decade or so, with the details depending on the amount of red noise in the pulsars. Detecting individual SMBBH systems is expected to shed light on the so-called final parsec problem, providing valuable insights into how SMBBHs merge. In fact, pulsar distances are well-measured, it may be possible to measure the SMBH spins using the pulsar terms [420].

Importantly, resolving an individual SMBBH with PTAs will open new avenues in multimessenger astronomy [421–424]. The combined GW and electromagnetic signals will allow to pin down the properties of the binary and its environment, to study the dynamics of the SMBBH pairing and the coupling with the surrounding gaseous disc, if present [362, 399]. Even a single joint detection will allow to nail down the distinctive electromagnetic properties of accreting SMBBHs. Analog systems can then be searched in the archival data of large surveys, to quantify their overall cosmic population. A promising way forward is to check candidates mined systematically in time domain surveys for peculiar spectral signatures, hence producing credible targets for PTAs.

Compact binary mergers (CBM) comprising at least one NS (i.e. BNS or BH-NS mergers) are unique cosmic laboratories for fundamental physics at the extreme. These powerful stellar collisions involve all fundamental interactions in a highly dynamical and intense regime (see, e.g. [425–428] and references therein for more comprehensive overviews). Their study is extremely challenging and requires multidimensional, multiphysics and multiscale models. General relativistic hydrodynamical (GRHD) simulations, including a detailed, microphysical description of matter and radiation, are necessary to model the merger and to produce robust predictions for a large variety of observables. Their comparison with observations will provide an unprecedented test for our understanding of the fundamental laws of Nature, in regimes that will never be accessible in terrestrial laboratories.

CBMs are events of primary relevance in astrophysics and fundamental physics. They are intense sources of neutrinos [429] and GWs [142], and primary targets for the present generation of ground-based GW detectors. Both theoretical and observational arguments support CBMs as progenitors of sGRB [430, 431]. At the same time, they are the places where the heaviest elements in the Universe (including Gold and Uranium) are synthesized and ejected into space, via the so-called r-process nucleosynthesis [432–434]. The radioactive decay of these freshly synthesized, neutron-rich elements in the ejecta powers a peculiar EM transient, called kilonova (or macronova), hours to days after the merger [435, 436]. Despite happening far away from the merger remnant, the $\gamma$-ray emission, the r-process nucleosynthesis, and the kilonova transient are extremely sensitive to physical processes happening where the gravitational curvature is stronger. In particular, the equation of state (EOS) of NS matter is thought to play a central role in all the emission processes, since it determines the compactness of the merging NSs, their tidal deformations, the lifetime and spin frequency of the remnant, the amount and the properties of the ejected mass.

On August, 17 2017, the first detection of GWs from a CBM event compatible with a BNS merger marked the beginning of the multimessenger astronomy era [21, 22] and remarkably confirmed several years of intense and productive lines of research. The GW signal, GW170817, provided the link to associate a sGRB, GRB170817A [23], and the kilonova AT2017gfo transient [180, 437, 438] to a CBM localized in the NGC4993 galaxy. Moreover, this single event set constraints on the EOS of NS matter and gave an independent measure of the Hubble constant [24].

Many relevant aspects of the merger and of the emission processes are, however, not yet fully understood and relate to open questions in fundamental physics and astrophysics. The interpretation of the observational data strongly relies on the theoretical modeling of compact binary sources in GR. During the last years the unprecedented growth of computing power has allowed the development of increasingly sophisticated numerical models. The most advanced simulations in Numerical Relativity (NR) are set up using a first-principles and an ab initio approach. Neutrino radiation and magnetic fields are thought to play a key role during the merger and its aftermath. Their inclusion in detailed NR simulations, in association with microphysical, finite temperature EOS for the description of matter at and above nuclear saturation density is one of the present challenges in the field. Reliable predictions do not only require the inclusion of all the relevant physics, but also accurate numerical schemes and numerically converging results. Robust and computationally stable discretizations are also essential to perform long-term and high-resolution simulations.

9.2.1. Gravitational waves and remnant properties.

9.2.2. Matter ejection and electromagnetic counterparts.

9.2.3. GW170817 and its counterparts.

Modelling of BH-NS mergers in GR shares many similarities with BNS merger modelling, but reveals also differences and peculiarities (see [426] for a review). The larger ranges in masses and spins expected for stellar mass BHs significantly increase the parameter space. The large mass ratio between the colliding objects makes NR simulations more expensive, since the inspiral requires larger initial distances to cover a sufficient number of orbits, and the dissimilar lengthscales require higher resolutions. These numerical challenges have limited the number of models and waveforms that are presently available in comparison with the BNS case. If the final remnant is always represented by a spinning BH, the presence of a disk around it depends on the mass ratio and BH spin [494–496]. Larger BHs, with moderate spins, swallow the NS during the plunge dynamical phase, while the decrease of the last stable orbit in the case of aligned, fast spinning BHs leads more easily to the tidal disruption of the NS (in the form of mass shedding) and to a massive accretion disk formation (with masses possibly in excess of 0.1 M$_\odot$). Misalignment between the orbital and the BH angular momentum induces non-trivial precessions, encoded both in the GW signal and in the amount of mass outside the BH horizon. The GW spectrum shows a characteristic cutoff frequency directly related with the orbital frequency at the NS tidal disruption or at the last stable orbit. Since this frequency decreases with increasing BH spin and with decreasing compactness, its detection could provide direct constraints on the NS matter EOS. Dynamical mass ejection from BH-NS merging system is expected to be caused by tidal torque when the NS is disrupted, and to happen predominantly along the equatorial plane. The ejected mass can be significantly larger than in the BNS case and neutrino irradiation is expected to have only a minor effect in changing the ejecta composition [497]. Similarly to the BNS case, the viscous evolution of the disk drives significant matter outflows on the disk lifetime, while the absence of a central NS reduces the emission of neutrinos and the ejection of $\nu$- or magnetically-driven winds [498]. The weaker effect of neutrino processes on the dynamic ejecta results always in robust r-process nucleosynthesis between the second and the third r-process peaks, while viscous disk winds can still synthesize all r-process nuclei. The high lanthanide content in the ejecta is expected to produce a redder kilonova transient a few days after the merger [499].

A complete and dense exploration of the wide parameter space, including a consistent treatment of NS and BH spins and of their evolution, is one of the major challenges in the study of CBMs in NR. The extraction of waveforms from long inspiral simulations at high resolution, employing accurate high-order numerical schemes, is necessary to build a robust NR waveform database. These templates, in combination with more analytical approaches, can guide the construction of complete and coherent semi-analytical waveforms for the GW data analysis, spanning many orbits from the inspiral to the actual coalescence. The large variety of properties in the colliding system potentially translates to a broad distribution of properties for the remnant and for the ejecta. Long term simulations of the merger and of its aftermath are the necessary tool to provide a complete and accurate description of the ejecta. The inclusion of the physics necessary to model the remnant and the matter ejection is still at its outset. In particular, the consistent inclusion of neutrino radiation and magnetic field evolution in NR model is extremely challenging. Leakage scheme for neutrino radiation are presently implemented in many CBM models (see, e.g. the introduction section of [500] for a detailed discussion). They provide a robust and physically motivated treatment for neutrino cooling. However, they are too inaccurate to model long term evolution and neutrino absorption in optically thin conditions. State-of-the-art simulations include gray moment scheme [463, 501]. Nevertheless, the significant dependence of neutrino cross-sections on particle energy requires spectral schemes. Large velocity gradients inside the computational domain make the accurate transformation of the neutrino energy spectrum between different observers and its transport highly non-trivial. Moreover, the application of moment schemes for colliding rays in free streaming conditions leads to closure artifacts in the funnel above the remnant. Monte Carlo radiation schemes represent an appealing alternative, but their computational cost is still far beyond our present capabilities. The usage of Monte Carlo techniques to provide more physical closures in moment scheme seems a more viable approach [502]. Neutrino masses have a potentially high impact on the propagation and flavor evolution of neutrinos from CBMs. New classes of neutrino oscillations, including the so-called matter neutrino resonances, can appear above the remnant of BNS and BH-NS mergers and influence the ejecta nucleosynthesis [503, 504]. However, numerical modelling of these phenomena is still at its dawn [476, 505, 506].

General relativistic magneto-hydrodynamics codes are presently available and they have started to access the spatial scales necessary to consistently resolve the magneto-rotational instabilities (MRIs, [507]). The latter appear in the presence of differential rotation and locally amplify the magnetic field. However, global simulations obtained with unprecedented spatial resolution (of the order of 12.5 m) do not show convergence yet, proving that we are still not able to resolve the most relevant scales for magnetic field amplification in a self-consistent way [508]. In the past, sub-grid models have been used as alternative approaches to ab initio treatments [454, 509, 510]. More recently, two different formulations of effective MHD-turbulent viscosity in General Relativity have been proposed and used in both global BNS and long term aftermath simulations [511, 512]. Moreover, the role of bulk viscosity in the post-merger phase has started being investigated [513].

Accurate modelling of the GW and EM signals for CBMs are key to set tight constraints on the EOS of NS matter, which still represents one of the major sources of uncertainty in both fundamental physics and in numerical models. If the connection between CBMs and short GRBs will be confirmed by future detections, the knowledge of the remnant fate, as well as of the environment around it, will be crucial to address the problem of short GRB engines, including jet formation, collimation, and break-out. The accurate modelling of small-scale magnetic field amplification, as well as of heat redistribution due to neutrino transport, is key to predict the lifetime of the remnant for cases of astrophysical interest. In fact, the presence of highly rotating and magnetized massive NSs [454, 514], or of fast spinning BHs, is anticipated to play an essential role in solving the puzzle of the short GRB central engine.

The high computational costs required by long-term, high-resolution, numerically accurate and multiphysics models of CBMs point to the need of developing a new generation of numerical schemes and codes for the new generations of large supercomputers. These codes will need, for example, to improve scalability and to employ more heavily vectorization in the hybrid (shared & distributed) parallelization paradigm. This is perhaps the greatest challenge in the NR field for the years to come.

For many years it was hypothesized that short gamma-ray bursts (GRBs) that are by now routinely observed, are originating from NS mergers (e.g. [530]). The theoretical model includes the launching of a relativistic jet during the short period of accretion onto the merger remnant. This energetic jet is believed to be responsible for the prompt gamma-ray emission. In addition, the interaction of the jet with the interstellar medium will produce an afterglow emission in the optical, x-ray and also radio wavelengths.

The short GRB ($T_{90} = 2.0 \pm 0.5$ s) that has been discovered about 1.7 s after the GW170817 merger event [531] was irregular compared to other short GRBs observed so far. Most notable is the low equivalent isotropic energy $E_{\rm iso} \approx 5 \times 10^{46}$ erg (in the 10–1000 kev band), which is  ∼4 orders of magnitude lower than a typical short GRB energy. In addition to the initial detection by Fermi-GBM, there were observations made by the Integral satellite. Integral also detected the short GRB with a flux of  ∼$1.4 \times 10^{-7}$ erg cm⁻² (at 75–2000 kev [532]).

Following the initial detection and once the candidate counterpart had been localized, x-ray observations were made within less than a day of the merger event. The initial x-ray observations by both the Neil Gehrels Swift Observatory and by the Nuclear Spectroscopic Telescope Array (NuSTAR) resulted in null-detections. However, later on, on day  ∼9 after the merger, the Chandra telescope detected x-ray emission from the AT 2017gfo for the first time [525] with an initial isotropic luminosity of $\approx$$9 \times 10^{38}$ erg s⁻¹. In the following days, the x-ray luminosity appeared to continue to slowly rise [525, 526]. Late-time observations (109 d after the merger) by Chandra still show that the x-ray emission continues to slowly rise [533].

In order to reconcile the relatively faint prompt gamma-ray emission and the late onset of the x-ray emission, it was suggested that AT 2017gfo was a regular short GRB but one that is observed slightly off the main axis by  ∼10 degrees of the jet [525]. However, this interpretation has yet to be tested against observations in other wavelengths, such as the radio (see below).

Over four decades ago, a prediction was made by [429, 432] that neutron-rich material can be tidally ejected in a NS merger. About a decade later numerical simulations also showed that NS mergers will exhibit such mass ejections, where the mass ejected is expected to be in the mass range 10⁻⁵–10⁻² M$_{\odot}$ with velocities of 0.1–0.3 c (e.g. [534]). It was also predicted that heavy elements would form in the neutron rich ejected material via r-processes. As discussed in detail in the previous section 9, additional material is also expected to be ejected by accretion disk winds and from the interface of the two merging stars (the latter material will be ejected mostly in the polar direction). In general, each ejected mass component may have a different neutron fraction leading to a different r-process element composition.

As the ejecta from the NS merger is radioactive, it can power transient emission, as proposed in [535]. The initial prediction was that the emission will be supernova-like and will be blue and peaking on a one-day timescale. Later on, more detailed calculations by [182] showed that the peak of the emission will be weaker at a level of 10⁴¹ erg s⁻¹. The next theoretical developments were made by [479, 480, 483] who for the first time calculated the effect of the heavy r-process element opacities on the emission. They found that due to very high opacities, the emission is expected to be even weaker, with its spectral peak now in the infrared (instead of the optical) and with the peak timescale being delayed. Since the various ejecta mass components may have different compositions, they therefore may have different opacities. Thus in principle, one component may form a blue short-lived emission (the so-called 'blue component') while another may emit in the infrared with week-long timescales (the so-called 'red component').

The optical emission from the AT 2017gfo peaked at a faint absolute magnitude of $M_{V} \approx -16$ in one day and began to rapidly decline, while the infrared emisson had a somewhat fainter and later (at approximately 2 d) peak compared to the optical emission. The overall observed brightness of the source in the optical and infrared bands and its evolution over time have shown in general an astonishing agreement with the predicted properties of such an event.

The multiple photometric and spectroscopic measurements sets obtained by the various groups paints the following picture: at early times, at about 0.5 d after the merger, the ejecta temperature was high at $T\sim 11\,000$ K (e.g. [520]). A day later the spectral peak was at $\approx$6000 $\mathring{\rm A}$, and the temperature decreased to $T\approx 5000$ K [437]. From this point onwards the spectral peak quickly moved into the infrared band. The combined optical and IR emission and its evolution were found to be consistent with an energy source powered by the radioactive decay of r-process elements (e.g. [438, 516, 517]). Based on both the photometric and spectroscopic analysis an estimate of the ejecta mass and its velocity were found to be ${\rm M}_{ej} \approx 0.02$–0.05 M$_{\odot}$ with a velocity in the range 0.1–0.3 c [437, 516–520, 522–524].

One of the main claims with regards to this event is that the observations provide evidence for the formation of r-process elements. This conclusion is mainly driven based on the evolution of the infrared emission [516, 521]. Tanvir et al [438], for example, obtained late-time IR measurements using the Hubble space telescope and argue that the slower evolution of the IR emission compared to the optical require high opacity r-process heavy elements with atomic number  >195. The infrared spectrum shows broad features which are presumably comprised of blended r-process elements [438]. These broad spectral features were compared to existing model predictions and show a general agreement, albeit there are still inconsistencies that need to be explained (e.g. [478, 521, 536]). There is still an ongoing debate about the composition of the heavier elements. [517] claims that the observed infrared spectral features can be matched with CS and Te lines. Others estimate the overall fraction of the lanthanides in the ejecta and find it to be in the range ${\rm X}_{lan} \approx 10^{-4}$–10⁻². It seems that at early times (up to 3–5 d), the ejecta that dominates the emission has very low lanthanide fraction and that there are discrepancies between the predicted and observed optical light curves (e.g. [519, 523]. Several works (e.g. [437, 478, 517, 518, 521, 524, 536, 537]) explain the early versus late-time behaviour of the emission by having an ejecta with two components (as also predicted in the literature). The first component is the 'blue' kilonova, with a high electron fraction and thus a low fraction of heavy elements, which dominates the optical emission early only. A 'red' kilonova is the second component that is comprised by heavy r-process elements and that produces the slow IR evolution and the broad spectral features observed at late-times. Still there are some claims that even at late times, the emission can be originating from an ejecta with only low-mass elements [538].

In addition to the radio afterglow emission on the short time scale (days), radio emission is also expected on longer time scales (month to years). The latter is not a result of the relativistic jet but rather originating from the interaction of the slower dynamical ejecta with the instersteIlar medium (ISM) [539]. This emission is expected to be rather weak where the strength of the peak emission depends on the velocity of the dynamical ejecta, the ISM density and on some microphysical parameters. While in the past, radio afterglows of short GRBs (not accompanied by GWs) were detected (e.g. [540]), long-term radio emission was never observed until GW 170817 (including in the previous cases where kilonova candidates were discovered [541, 542]).

Similar to any other wavelength, radio observations were undertaken within the day of the GW170817 merger discovery and the following days. The early time observations performed by The Jansky Very Large Array (VLA), The Australian Compact Array (ATCA), the Giant Meterwave Radio Telescope (GMRT) and the Atacama Large Millimeter Array (ALMA), all resulted in null-detections [528, 543, 544]. Late-time VLA observations, however, finally revealed a radio counterpart $\approx$16 d after the merger [528]. The initial radio emission was weak at a level of a couple of tens of $\mu$Jy only (at both 3 and 6 GHz). Follow-up ATCA observations confirmed the radio detection. Upon detection, a long-term radio monitoring campaign was initiated, and the results are reported in [529]. They report that the radio emission still continued to rise at  >100 d after the merger. The multiple frequency observations also show that the radio emission is optically thin with a spectral index of $\alpha = -0.6$.

Hallinan et al [528] compared the radio observations to several predictions including an on-axis jet, a slightly off-axis jet, and being completely off-axis. In addition, they included in their comparison a model in which the jet forms a hot wide-angled mildly relativistic cocoon. This cocoon formed as the jet is working its way out of the dynamical ejecta, may also lead to gamma-ray, x-ray, and radio emission, but with different characteristics than the emission formed by either the highly relativistic jet or the slower dynamical ejecta. In fact, [528] find that both an on-axis jet model, or even a slightly off-axis jet one, are expected to produce bright radio emission in the first few days after the merger which by day 16 should start fading, a prediction which does not match the rising observed radio source. The model that is the most consistent with current data is the cocoon model with either a choked or successful or 'structured' jet (see e.g. [533, 545]). In fact, the choked-jet cocoon model may also explain the relatively low energy of the gamma ray emission [546].

If both the late-time radio and x-ray emission originate from interaction of the material (whether it is the dynamical ejecta or a cocoon) with the ISM, one can test whether the observed emission in both wavelengths is indeed connected as expected. For now, it seems that the observed x-ray emission fits the prediction which is based on extrapolating the observed radio emission into higher energies. Both the x-rays and the radio emission have also roughly the same spectral slope. This suggests that there is no additional power source in play at this time.

While vast amounts of data have been collected for this amazing merger event, and while a flood of papers report many types of analysis and conclusions (by no means are we attempting to cover all of them here), there are still many open questions remaining. For example, is this event connected to short GRBs or do we still have to prove this connection? Which r-process elements form and at what time? Do the observations really require producing heavy r-process elements or can they all be explained by lighter-element components? How many components does the ejecta have and which one dominates the emission and when? Are there any other emission power sources in play such as a magnetar (even if at short time)? Are we really seeing a cocoon with a choked or successful jet or is there some other scenario that can explain the radio emission combined with all the other evidence?

As scientists around the world are still working on analyzing all the data in hand for this event and are also still collecting new data, they are also gearing up for the future, and preparing for the next year when the LIGO and Virgo detectors switch back on. At this time, hopefully more events with EM signatures will be discovered and more answers (and surely more new questions) will present themselves.

Although the identification of compact SMBBHs might have an important impact on GW observations with PTAs in the near future, looking further ahead, LISA is expected to detect tens to hundreds of coalescing SMBBHs throughout the Universe. Electromagnetic observations related to the merger of SMBBHs are important both for cosmology and the astrophysics of galactic nuclei. Pinning down the host galaxy of the merger will allow us to combine redshift and distance from gravitational waves to constrain cosmological parameters ([609], see next section) and to study the large scale galactic environment of merging SMBBHs, adding to our understanding to the process galaxy formation. On the other hand, electromagnetic observations will give us access to the properties of matter in the relative close environment of a merger and to the gas an stars (hydro)-dynamics as they adjust in response to the merger.

Given the importance of such identification, there has been an extensive effort to predict observable electromagnetic signatures that can occur in nearly coincidence with the event ('prompt signals') or afterwards ('afterglows'). In the following few examples will be given; notably, some of them also inspired recent models for possible electromagnetic counterparts to SOBBH mergers, of the kind detected by LIGO and VIRGO. As mentioned previously, SMBBH mergers, especially at high redshifts, can happen in a gaseous environment that provides each SMBH with a 'minidisc', fed through streams leaking from a circumbinary disc. Those minidiscs are likely to be retained even after the orbital decay due to gravitational radiation dominates, providing distinctive modulation of the emerging luminosity as the binary spirals in [610, 611]. During the final orbits, the surviving gas between the black holes gets squeezed possibly producing super-Eddington outflows as discussed in [612, 613], but see [614]. Full GR simulations have also been employed to study the possible formation of precessing jets during the inspiral and merger, in the attempt of identifying distinctive signatures [615–617]. General relativity predicts that a newly formed black hole suffers a recoil because GWs carry away a non-zero linear momentum (e.g. [618, 619]). This recoil affects the circumbinary disc, bound to the black hole: a kick is imparted that shocks the gas producing a slowly rising, ∼10 yr lasting afterglows [620–624]. In stellar mass black hole merger, similar phenomena but on much shorter timescales can occur [171, 625]. Contrary to the SMBH case, however, providing a gas rich environment for the merger is a challenge. A possible venue involves cold relic discs, formed as a result of weak supernovae, where accretion is suppressed until either the 'squeezing' or the 'kicking' heat them up again in the same configuration envisaged for SMBHs.

As already mentioned, the next theoretical challenge for these dynamical models is to predict realistic lightcurves and spectra, which will require non-trivial radiative transfer calculations (see, e.g. [626]). With solid predictions in hand, appropriate strategies can be devised to coordinate electromagnetic follow-ups, to take full advantage of multimessenger astronomy in the LISA era.

Within the theory of GR, a binary system composed by two compact astrophysical objects orbiting around each other, emits a GW signal with the two polarizations [627]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle h_\times(t) = \frac{4}{d_L(z)} \left( \frac{G \mathcal{M}_c(z)}{c^2} \right)^{5/3} \left( \frac{\pi f(t)}{c} \right)^{2/3} \sin[\Phi(t)] \cos\iota, \label{eq:hx} \nonumber \end{align} \tag{ 8 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle h_+(t) = h_\times(t) \frac{1+\cos^2 \iota}{2 \cos\iota} \cot[\Phi(t)] , \label{eq:h+} \nonumber \end{align} \tag{ 9 }$$

where $h_{\times,+}(t)$ are the GW strains in the transverse-traceless gauge (we neglect here post-Newtonian contributions). In these expressions $\iota$ is the orientation of the orbital plane with respect to the detector, z is the redshift of the source, d_L(z) is the luminosity distance, f  is the GW frequency at the observer, $\Phi(t)$ is the phase of the GW, and $\mathcal{M}_c(z)=(1+z)(m_1 m_2){}^{3/5}/(m_1+m_2){}^{1/5}$ is the redshifted chirp mass, with m_1,2 being the masses of the two binary bodies. An accurate detection of the GW signal allows to reconstruct all the parameters in equations (8) and (9) within some (correlated) uncertainties [628]. In particular, thanks to the reconstruction of d_L, binaries can be employed as reliable cosmological distance indicators.

Equations (8) and (9) highlight three key aspects of using inspiralling binaries as cosmological distance indicators: (i) The measurement of d_L from GW signals is not affected by any systematic uncertainties in the modelling of the source, since the dynamics of compact binary systems is directly determined by GR. This is in contrast with supernovae type-Ia (SNIa), which require the cosmic distance ladder, i.e. cross calibration with local measurements of sources of known distance, to overcome unknown systematics in the determination of their luminosity distance. (ii) Due to the scaling $\propto$$d_L^{-1}$, GW cosmological indicators are suitable even at large distances where EM sources, whose intensity scales as $d_L^{-2}$, are too faint. This implies that given the same amount of emitted energy, a source producing GWs can be observed at higher distances with respect to a source emitting EM waves. (iii) The measurement of the quoted waveform does not allow to determine the redshift of the source: in fact equations (9) and (8) are invariant under the transformation $m_i\to m_i (1+z)$ plus $d_L \to d_L (1+z)$. In other words the waveform detected from any system with masses m_i at a distance d_L will be equivalent to a waveform produced by a system with masses m_i(1  +  z) at a distance d_L(1  +  z).

The luminosity distance d_L is tightly linked to the redshift z in a given cosmological setup. For a homogeneous and isotropic universe (at large scale), the luminosity distance is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle d_L(z) = \frac{c}{H_0}\frac{1+z}{\sqrt{\Omega_k}} \sinh \left[ \sqrt{\Omega_k} \int_0^z \frac{H_0}{H(z')} {\rm d}z' \right] , \label{eq:dL_z} \nonumber \end{align} \tag{ 10 }$$

with $\Omega_k$ being the present value of the density parameter of the spatial curvature, $H(z)$ being the Hubble rate as a function of the redshift, and H₀  =  H(z  =  0). Therefore, if besides d_L (reconstructed from the waveform) one can independently establish the redshift z of a GW source, then one obtains a data point useful to constrain the relation (10). The parameters of any given cosmological model, which are implicitly included in $H(z)$ since its dynamics is determined by the Einstein equations, can then be statistically constrained by fitting equation (10) against a number of (z, d_L(z)) data points. This can be done using observations from GW inspiralling binaries, if a determination of their redshift is available by some means. For this reason, well detectable binaries, for which the redshift is also known (or at least estimated), are dubbed standard sirens, in analogy with SNIa used as standard candles in cosmography [629].

Of course, to obtain a robust bound, it is crucial to fit equation (10) with as many standard sirens as possible at the most diverse redshifts, especially if the cosmological model at hand contains many parameters. The outcome of such an analysis can be remarkable. It constitutes the first robust cosmological test not using EM radiation as the only messenger of astronomical information, and it allows to probe the validity of the cosmic distance ladder up to far distances. As demonstrated by the recent analysis of the LVC detection GW170817, discussed in section 13.1.2 below, it also provides a measurement of H₀ that is independent of the calibrations necessary to establish constraints using SNIa. In particular, if the current tension between the H₀ measurement from SNIa and CMB (assuming $\Lambda$CDM) will persist, standard sirens will be a very useful observable to decipher the origin of this tension.

13.1.1. Redshift information.

13.1.2. Standard sirens with current GW data: GW170817.

13.1.3. Cosmological forecasts with standard sirens.

13.1.4. Future prospects.

The gravitational interaction is so weak that GWs propagate practically unperturbed along their path from the source to us. GWs produced in the early universe thus can carry a unique imprint of the pre-CMB era, in which the universe was not transparent to photons. In this sense, GW detection can provide for the first time direct, clean access to epochs that are very hard to probe by any other observational means. GWs of cosmological origin appear (pretty much as the CMB) to our detectors as a SGWB [684, 685].

Several pre-CMB phenomena sourcing GWs might have occurred along the cosmological history, from inflation to the epoch of the QCD phase transition. In such a case, the SGWB would be constituted by the sum of all single contributions, each of them potentially differing from the others in its spectral shape as a function of frequency, or because of other properties such as chirality and/or gaussianity. It is hence crucial to have precise predictions of all proposed cosmological sources to possibly isolate all components in the detected SGWB, with the aim of reconstructing the early time history of the universe.

On the other hand, binary systems can also be detected as a SGWB. This happens for those (independent) astrophysical events that are overall too weak to be individually resolved. The level of their 'contamination' to the cosmological SGWB thus depends on the sensitivity and on the resolution of the available detector. Crucially, at LIGO-like and LISA-like experiments the astrophysical component might be stronger than the cosmological one, so that the information hidden in the pre-CMB signal would be impossible to recover, unless the astrophysical contribution is known in great detail and methods are found to subtract it.

13.2.1. Status and future prospects.

Focused work on the GSF problem has been ongoing for at least the last two decades, during which time there has been substantial progress. Early work developed much of the mathematical formalism, particularly in how one constructs a well-motivated and unambiguous regularized first-order metric perturbation [717–723, 735]. More recent work has addressed the conceptual challenges around how these initial results can be extended to second perturbative order [725–731], and on turning the formal mathematical prescriptions into practical numerical schemes [736–741]. As a result, we are now at the point where first-order GSF calculations are possible for almost any orbital configuration in a Kerr background spacetime [742]. Indeed, we now have not one, but three practical schemes for computing the regularized first-order GSF [743]. Some of the most recent highlights from the substantial body of work created by the GSF community are discussed below.

2.1.1. First-order gravitational self-force for generic orbits in Kerr spacetime.

2.1.2. Extraction of gauge-invariant information.

2.1.3. Synergy with PN approximations, EOB theory, and NR.

2.1.4. New and efficient calculational approaches.

2.1.5. Cosmic censorship.

While the perturbative (self-force) approach has proven highly effective to date, there remain several important and challenging areas for further development. Here, we list a number of the most important future challenges and prospects for the GSF programme.

2.2.1. Efficient incorporation of self-force information into waveform models.

2.2.2. Producing accurate waveform models: self-consistent evolution and second-order gravitational self-force.

2.2.3. Gravitational Green function.

2.2.4. Internal-structure effects.

2.2.5. EMRIs in alternative theories of gravity.

2.2.6. Open tools and datasets.

The PN formalism is an approximation method in GR that is well suited to describe the orbital motion and the GW emission from binary systems of compact objects, in a regime where the orbital velocity is small compared to the speed of light and the gravitational fields are weak. This approximation method has played a key role in the recent detections, by the LIGO and Virgo observatories, of GWs generated by inspiralling and merging BH and NS binaries [1, 2, 11, 18, 21], by providing accurate template waveforms to search for those signals and to interpret them. Here we give a brief overview of the application of the PN approximation to binary systems of compact objects, focusing on recent developments and future prospects. See the review articles [906–913] and the textbooks [627, 914] for more information.

In PN theory, relativistic corrections to the Newtonian solution are incorporated in a systematic manner into the equations of motion (EOM) and the radiation field, order by order in the small parameter $v^2/c^2 \sim Gm/(c^2r)$, where $v$ and r are the typical relative orbital velocity and binary separation, m is the sum of the component masses, and we used the fact that $v^2 \sim Gm/r$ for bound motion. (The most promising sources for current and future GW detectors are bound systems of compact objects.)

Another important approximation method is the post-Minkowskian (PM) approximation, or non-linearity expansion in Newton's gravitational constant G, which assumes weak fields ($Gm/c^2 r \ll 1$) but unrestricted speeds ($v^2 / c^2 \lesssim 1$), and perturbs about the limit of special relativity. In fact, the construction of accurate gravitational waveforms for inspiralling compact binaries requires a combination of PN and PM techniques in order to solve two coupled problems, namely the problem of motion and that of wave generation.

The two-body EOM have been derived in a PN framework using three well-developed sets of techniques in classical GR: (i) the PN iteration of the Einstein field equations in harmonic coordinates [915–919], (ii) the Arnowitt–Deser–Misner (ADM) canonical Hamiltonian formalism [920–922], and (iii) a surface integral approach pioneered by Einstein, Infeld and Hoffmann [923–925]. By the early 2000s, each of these approaches has independently produced a computation of the EOM for binary systems of non-spinning compact objects through the 3rd PN order (3PN)¹³⁶. More recently, the application of effective field theory (EFT) methods [926], inspired from quantum field theory, has provided an additional independent derivation of the 3PN EOM [927]. All of those results were shown to be in perfect agreement. Moreover, the 3.5PN terms—which constitute a 1PN relative correction to the leading radiation-reaction force—are also known [928–933].

At the same time, the problem of radiation (i.e. computing the field in the far/wave zone) has been extensively investigated within the multipolar PM wave generation formalism of Blanchet and Damour [934–936], using the 'direct integration of the relaxed Einstein equation' approach of Will, Wiseman and Pati [937, 938], and more recently with EFT techniques [939, 940]. The application of these formalisms to non-spinning compact binaries has, so far, resulted in the computation of the GW phase up to the relative 3.5PN (resp. 3PN) order for quasi-circular (resp. quasi-eccentric) orbits [941–948], while amplitude corrections in the GW polarizations are known to 3PN order, and even to 3.5PN order for the quadrupolar mode [949–954].

Over the last five years or so, significant progress on PN modelling of compact binary systems has been achieved on multiple fronts, including (i) the extension of the EOM to 4PN order for non-spinning bodies (with partial results also obtained for aspects of the two-body dynamics at the 5PN order [814, 823, 955]), (ii) the inclusion of spin effects in the binary dynamics and waveform, (iii) the comparison of several PN predictions to those from GSF theory, and (iv) the derivation of general laws controlling the mechanics of compact binaries.

3.2.1. 4PN equations of motion for non-spinning compact-object binaries.

3.2.2. Spin effects in the binary dynamics and gravitational waveform.

Since stellar-mass and/or supermassive BHs may carry significant spins [975, 976], much effort has recently been devoted to include spin effects in PN template waveforms. In particular, spin–orbit coupling terms linear in either of the two spins have been computed up to the next-to-next-to leading order, corresponding to 3.5PN order in the EOM, using the ADM Hamiltonian framework [977, 978], the PN iteration of the Einstein field equations in harmonic coordinates [979, 980], and EFT techniques [981]. Spin-spin coupling terms proportional to the product of the two spins have also been computed to the next-to-next-to leading order, corresponding to 4PN order in the EOM, using the ADM Hamiltonian and EFT formalisms [982–986]. The leading order 3.5PN cubic-in-spin and 4PN quartic-in-spin contributions to the binary dynamics are also known for generic compact bodies [987–991], as well as all higher-order-in-spin contributions for BBHs (to leading PN order) [992]. All these results are summarized in figure 8. 2PN BH binary spin precession was recently revisited using multi-timescale methods [993, 994], uncovering new phenomenology such as precessional instabilities [995] and nutational resonances [996].

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Spin-related effects on the far-zone field have also been computed to high orders, for compact binaries on quasi-circular orbits. To linear order in the spins, those effects are known up to the relative 4PN order in the GW energy flux and phasing [997, 998], and to 2PN in the wave polarizations [999, 1000]. At quadratic order in the spins, the contributions to the GW energy flux and phasing have been computed to 3PN order [1001, 1002], and partial results were derived for amplitude corrections to 2.5PN order [1003]. The leading 3.5PN cubic-in-spin effects in the GW energy flux and phasing are known as well [989].

Finally, some recent works have uncovered remarkable relationships between the PN [992] and PM [1004] dynamics of a binary system of spinning BHs with an arbritrary mass ratio on the one hand, and that of a test BH in a Kerr background spacetime on the other hand. Those results are especially relevant for the ongoing development of EOB models for spinning BH binaries (see section 6), and in fact give new insight into the energy map at the core of such models.

3.2.3. Comparisons to perturbative gravitational self-force calculations.

3.2.4. First law of compact binary mechanics.

On the theoretical side, an important goal is to extend the knowledge of the GW phase to the relative 4.5PN accuracy, at least for non-spinning binaries on quasi-circular orbits. (Partial results for some specific tail-related effects were recently derived [1020].) This is essential in order to keep model systematics as a sub-dominant source of error when processing observed GW signals [1021]. Accomplishing that will require, in particular, the calculation of the mass-type quadrupole moment of the binary to 4PN order and the current-type quadrupole and mass-type octupole moments to 3PN order. Moreover, some spin contributions to the waveform—both in the phasing and amplitude corrections—still have to be computed to reach the 4PN level, especially at quadratic order in the spins.

Most of the PN results reviewed here have been established for circularized binaries, and often for spins aligned or anti-aligned with the orbital angular momentum. It is important to extend this large body of work to generic, eccentric, precessing systems. Progress on the two-body scattering problem would also be desirable [1022–1024]. Additionally, much of what has been achieved in the context of GR could be done as well for well-motivated alternative theories of gravitation, such as for scalar-tensor gravity (e.g. [1025–1029]) or in quadratic gravity [1030, 1031].

The first law of binary mechanics reviewed above could be extended to generic, precessing spinning systems, and the effects of higher-order spin-induced multipoles should be investigated. Recent work on the PM approximation applied to the gravitational dynamics of compact binaries has given new insight into the EOB model [1004, 1032–1034], and in particular into the energy map therein. These two lines of research may improve our physical understanding of the general relativistic two-body problem.

On the observational side, future GW detections from inspiralling compact-object binaries will allow testing GR in the strong-field/radiative regime, by constraining possible deviations from their GR values of the various PN coefficients that appear in the expression of the phase. This, in particular, can be used to test some important nonlinear features of GR such as the GW tail, tail-of-tail and nonlinear memory effects. Indeed, the first detections of GWs from inspiralling BH binaries have already been used to set bounds on these PN coefficients, including an $O(10\%)$ constraint on the leading tail effect at the 1.5PN order [3]; see figure 9 (or [18] for a more up-to-date version thereof). More detections with a wider network of increasingly senstive interferometric GW detectors will of course improve those bounds.

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Finally, following the official selection of the LISA mission by the European Space Agency (ESA), with a launch planned for 2034, we foresee an increased level of activity in source modelling of binary systems of MBHs and EMRIs, two promising classes of sources for a mHz GW antenna in space. This will motivate more work at the interface between the PN approximation and GSF theory.

4.1.1. Excision and puncture.

4.1.2. Initial data.

4.1.3. Evolution systems.

4.1.4. Gravitational wave extraction.

4.2.1. High spins.

4.2.2. High mass ratios.

4.2.3. Long simulations.

4.2.4. Waveform accuracy.

4.2.5. Beyond general relativity.

NR plays an important role in GW astrophysics and data analysis. The semi-analytical waveform models employed to search for BBHs (see section 6 below) model the complete radiative evolution from the early inspiral, through the merger and to the final ringdown stage, with the later stages calibrated to BBH simulations [1107, 1108]. These models and more sophisticated ones incorporating more physical effects, for example precession [1109, 1110], are used to determine the fundamental properties of the GW source, i.e. its masses and spin angular momenta as well as extrinsic quantities such as the orbital orientation relative to the observatories [1111, 1112].

Alternatively, pure NR waveforms may be used directly [1113, 1114] or by the means of surrogate models [1115–1117]. But due to the computational cost of simulations, these have only been attempted so far on a very restricted portion of the parameter space. Despite this restriction, pure NR surrogate models have the advantage of incorporating more physical effects that may be limited or entirely neglected in currently available semi-analytic models.

The ringdown phase after the merger may be described by perturbation theory (see e.g. [1118] for a review), but the amplitudes of the excited quasi-normal modes can only be obtained from numerical simulations. These are of particular interest as measuring the amplitudes of individual quasinormal modes would allow to map the final state of the merger to the properties of the progenitor BHs [1119, 1120] and to test the BH nature of the source [1121] (see also sections 4.1 and 4.2).

In order to estimate the mass and spin angular momentum of the remnant BH, fits to NR simulations are essential [1088–1090, 1122–1129]. Independent measurements of the binary properties from the inspiral portion of the GW signal and from the later stages of the binary evolution using fits from NR allow, when combined, to test the predictions from GR [17, 1130]. While the phenomenological fit formulae have seen much improvement in recent years, modelling the final state of precessing BH binaries from numerical simulations still remains an open challenge.

Generically, when BHs merge, anisotropic emission of GWs leads to the build up of a net linear momentum. Due to the conservation of momentum, once the GW emission subsides, the remnant BH recoils ('kick'). While the recoil builds up during the entire binary evolution, it is largest during the non-linear merger phase. Kick velocities can be as high as several thousands ${\rm km}/{\rm s}$, with astrophysical consequences: a BH whose recoil velocity is larger than the escape velocity of its galaxy may leave its host. Numerical simulations are necessary to predict the recoil velocities [1131–1134] (a convenient surrogate model for these velocities was constructed recently in [1135]). It has been suggested that it may be possible to directly measure the recoil speed from the GW signal by observing the induced differential Doppler shift throughout the inspiral, merger and ringdown [1136].

Numerical simulations of the strong field regime are also crucial for exploring other spin-related phenomena such as 'spin-flips' [1137] caused by spin–spin coupling, whose signatures may be observable. Understanding the spin evolution and correctly modelling spin effects is crucial for mapping out the spin distribution of astrophysical BHs from GW observations.

Waveform models as well as fitting formulae for remnant properties are prone to systematic modelling errors. For GW observations with low SNR, the statistical uncertainty dominates over the systematic modelling error in the parameter measurement accuracy. Improved sensitivities for current and future GW observatories (including, especially, LISA [1138, 1139]) will allow for high SNR observations reaching into the regime where systematic errors are accuracy limiting. This includes strongly inclined systems, higher-order modes, eccentricity, precession and kicks, all of which can be modelled more accurately through the inclusion of results from NR (see [1021] for a detailed discussion in the context of the first BBH observation GW150914).

DM, by its very definition, generates little if any electromagnetic radiation; rather, it interacts with its environment through gravity. Many DM candidates have been suggested (see also section 5), ranging from primordial BH clouds to weakly interacting massive particles (WIMPs) and ultralight bosonic fields [281, 282, 1142]. The latter are a particularly attractive candidate in the context of BH and GW physics due to their specific interaction with BHs. Ultralight fields, also referred to as weakly interacting slim particles (WISPs), typically have mass parameters orders of magnitude below an electron volt and arise in extensions of the SM of particle physics or extra dimensions in string theory. These include axions or axion-like particles, dark photons, non-topological solitons (so-called Q-balls) and condensations of bosonic states [281, 1143–1145]. For reference, we note that a mass $10^{-10}~{\rm eV}$ ($10^{-20}~{\rm eV}$) corresponds to a Compton wavelength of $\mathcal{O}(1)~{\rm km}$ ($\mathcal{O}(10^2)~{\rm AU}$) which covers the range of Schwarzschild radii of astrophysical BHs.

At first glance, one might think the interaction between such fundamental fields and BHs is simple; stationary states are constrained by the no-hair theorems and the field either falls into the BH or disperses away. In practice, however, the situation is more complex—and more exciting. NR simulations have illustrated the existence of long-lived, nearly periodic states of single, massive scalar or Proca (i.e. vector) fields around BHs [1146–1148]. Intriguingly, these configurations are able to extract rotational energy from spinning BHs through superradiance [1149–1152], an effect akin to the Penrose process [1152, 1153]. Further details on this process are provided in section 5.7. In the presence of a confining mechanism that prevents the field from escaping to infinity, this may even lead to a runaway instability dubbed the BH bomb [1154, 1155]. Another peculiar consequence arising in the same context is the possibility of floating orbits where dissipation of energy through GW emission is compensated by energy gain through superradiance [1156, 1157].

Naturally, non-linear effects will limit the growth of the field amplitude or the lifetime of floating orbits, and recent years have seen the first numerical studies to explore the role of non-linearities in superradiance. These simulations have shown that massive, real scalar fields around BHs can become trapped inside a potential barrier outside the horizon, form a bound state and may grow due to superradiance [1146]. Furthermore, beating phenomena result in a more complex structure in the evolution of the scalar field. These findings were confirmed in [1147], which also demonstrated that the scalar clouds can source GW emission over long timescales. The amplification of GWs through superradiance around a BH spinning close to extremality has been modeled in [1158] and found to be maximal if the peak frequency of the wave is above the superradiant threshold but close to the dominant quasi-normal mode frequency of the BH. The generation of GW templates for the inspiral and merger of hairy BHs and the identification of possible smoking gun effects distinguishing them from their vacuum counterparts remains a key challenge for future NR simulations.

Numerical studies of the non-linear saturation of superradiance are very challenging due to the long time scales involved. A particularly convenient example of superradiance in spherical symmetry arises through the interaction of charged, massless scalar fields with Reissner–Nordström BHs in asymptotically anti-de Sitter spacetime or in a cavity; here energy is extracted from the BH charge rather than its rotation. The scalar field initially grows in accordance with superradiance, but eventually saturates, leaving behind a stable hairy BH [1159, 1160]. Recently the instability of spinning BHs in AdS backgrounds was studied in full generality [1161]. The system displays extremely rich dynamics and it is unclear if there is a stationary final state.

The superradiant growth of a complex, massive vector field around a near-extremal Kerr BH has recently been modeled in axisymmetry [1162]. Over $9\%$ of the BH mass can be extracted until the process gradually saturates. Massive scalar fields can also pile up at the center of 'normal' stars. Due to gravitational cooling, this pile-up does not lead to BH formation but stable configurations composed of the star with a 'breathing' scalar field [1163].

The hairy BH configurations considered so far are long-lived but not strictly stationary. A class of genuinely stationary hairy BH spacetimes with scalar or vector fields has been identified in a series of papers [1164–1168]. The main characteristic of these systems is that the scalar field is not stationary—thus bypassing the no-hair theorems—but the spacetime metric and energy-momentum are. A subclass of these solutions smoothly interpolates between the Kerr metric and boson stars. A major challenge for numerical explorations is to evaluate whether these solutions are non-linearly stable and might thus be astrophysically viable. A recent study of linearized perturbations around the hairy BH solution of [1164] has found unstable modes with characteristic growth rates similar to or larger than those of a massive scalar field on a fixed Kerr background. However, such solution may still be of some astrophysical relevance [1169]. See also section 4.2.1 of chapter III for a related discussion.

The idea of stationary localized, soliton-like configurations made up of the type of fundamental fields discussed in the previous section goes back to Wheeler's 'gravitational-electromagnetic entities' or geons of the 1950s [1170]. While Wheeler's solutions turn out to be unstable, replacing the electromagnetic field with a complex scalar field leads to 'Klein–Gordon geons' first discovered by Kaup [1171] and now more commonly referred to as boson stars. The simplest type of boson stars, i.e. stationary solutions to the Einstein-complex-Klein–Gordon system, is obtained for a non-self-interacting field with harmonic time dependence $\phi \propto {\rm e}^{{\rm i}\omega t}$ where the potential contains only a mass term $V(\phi)= m^2 |\phi|^2$. The resulting one-parameter family of these so-called mini boson star solutions is characterized by the central amplitude of the scalar field and leads to a mass-radius diagram qualitatively similar to that of static NSs; a maximum mass value of $M_{\rm max}=0.633\,M_{\rm Planck}^2/m$ separates the stable und unstable branches [1172–1174]. For a particle mass $m=30\,{\rm GeV}$, for instance, one obtains $M_{\rm max}\sim 10^{10}\,{\rm kg}$ with radius $R\sim 6\times 10^{-18}$ m and a density 10⁴⁸ times that of a NS [1175].

A wider range of boson star models is obtained by adding self-interaction terms to the potential $V(\phi)$ which result in more astrophysically relevant bulk properties of the stars. For a quartic term $\lambda |\phi|^4/4$, for example, the maximal mass is given by $M_{\rm Planck} (0.1\,{\rm GeV}^2)\,M_{\odot}\,\lambda^{1/2}/m^2$ [1176]; for further types of boson stars with different potential terms see the reviews [1174, 1175, 1177] and references therein. A particularly intriguing feature of rotating boson stars exemplifies their macroscopic quantum-like nature: the ratio of angular momentum to the conserved particle number must be of integer value which prevents a continuous transition from rotating to non-rotating configurations [1175, 1178]. More recently, stationary, soliton-like configurations have also been found for complex, massive Proca fields [1179]. For real scalar fields, in contrast, stationary solutions do not exist, but localized, periodically oscillating solutions dubbed oscillatons have been identified in [1180].

Boson stars are natural candidates for DM [1174], but may also act as BH mimickers [1181, 1182]. In the new era of GW observations, it is vital to understand the GW generation in boson-star binaries and search for specific signatures that may enable us to distinguish them from BH or NS systems. Recent perturbative calculations of the tidal deformation of boson stars demonstrate that the inspiral part of the waveform may allow us to discriminate boson stars, at least with third-generation detectors [1183]. Numerical studies of dynamic boson stars have so far mostly focused on the stability properties of single stars and confirmed the stable and unstable nature of the branches either side of the maximum mass configuration; see e.g. [1184–1187]. The modeling of head-on collisions of boson stars [1188, 1189] reveals rich structure in the scalar radiation and that the merger leads to the formation of another boson star. Head-on collisions have also served as a testbed for confirming the validity of the hoop conjecture in high-energy collisions [1190]. Inspiralling configurations result either in BH formation, dispersion of the scalar field to infinity or non-rotating stars [1191, 1192], possibly a consequence of the quantized nature of the angular momentum that makes it difficult to form spinning boson stars instead.

Binary boson star systems thus remain largely uncharted territory, especially regarding the calculation of waveform templates for use in GW data analysis and the quantized nature of spinning boson stars and their potential constraints on forming rotating stars through mergers.

New physical phenomena and the signature of new 'modified' theories are typically encountered when probing extreme regimes not accessible to previous experiments and observations. Quantum effects, for instance, become prominent on microscopic scales and their observation led to the formulation of quantum theory while classical physics still provides an accurate description of macroscopic systems. Likewise, Galilean invariance and Newtonian theory accurately describe slow motion and weakly gravitating systems but break down at velocities comparable to the speed of light or in the regime of strong gravity. We therefore expect modifications of GR, if present, to reveal themselves in the study of extreme scales such as the large-scale dynamics of the universe or the strong curvature regime near compact objects.

The dawn of GW observations provides us with unprecedented opportunities to probe such effects. The modeling of compact objects in alternative theories of gravity represents one of the most important challenges for present and future NR studies, so that theoretical predictions can be confronted with observations. This challenge faces additional mathematical, numerical and conceptual challenges as compared to the GR case; a more detailed discussion of these is given in section 3.4 below. A convenient way to classify the numerous modified theories of gravity is provided by the assumptions underlying Lovelock's theorem and, more specifically, which of these assumptions are dropped [1193]. Unfortunately, for most of these candidate theories, well-posed formulations are not known (see table 1 in [1193]) or presently available only in the form of a continuous limit to GR or linearization around some background [1104, 1106]. Prominent exceptions are (single- or multi-) scalar-tensor (ST) theories of gravity [1194] which includes Brans–Dicke theory [1195] and, through mathematical equivalence, a subset of $f(R)$ theories [1196]. ST theories inherit the well-posedness of GR through the Einstein frame formulation [1197]; see also [1198–1200]. Furthermore, ST physics would present the most conspicuous strong-field deviation from GR discovered so far, the spontaneous scalarization of NSs [1201] (for a similar effect in vector-tensor theory see [1202]). For these reasons, almost all NR studies have focused on this class of theories, even though its parameter space is significantly constrained by solar system tests and binary pulsar observations [1203].

The structure of equilibrium models of NSs in ST theories has been studied extensively in the literature (e.g. [1201, 1204–1210]) and leads to a mass-radius diagram that contains one branch of GR or GR-like models plus possible additional branches of strongly scalarized stars. The presence or absence of these additional branches depends on the coupling between the scalar and tensor sector of the theory.

Early numerical studies considered the collapse of dust in spherical symmetry [1211–1214] which leads to a hairless BH in agreement with the no-hair theorems, even though departures from GR are possible during the dynamic stages of the collapse. In a sequence of papers, Novak et al studied the collapse of NSs into a BH [1215], the transition of NSs between stable branches [1216] and the formation of NSs through gravitational collapse [1217]. In all cases, strong scalar radiation is generated for that part of the ST theory's parameter space that admits spontaneously scalarized equilibrium models. In [1218], the collapse of stellar cores to BHs was found to be the most promising scenario to generate detectable scalar radiation for parameters allowed by the Cassini and pulsar observations; galactic sources at a distance $D=10\,{\rm kpc}$ may be detected with present and future GW observatories or used to further constrain the theory's parameters. All these simulations, however, consider massless ST theory. For massive fields, low frequency interactions decay exponentially with distance, so that the pulsar and Cassini constraints may no longer apply [1219]. In consequence, massive ST theory still allows for very strongly scalarized equilibrium stars if $m\gtrsim 10^{-15}~{\rm eV}$ [1209, 1210]. This has dramatic consequences for the GW signals that can be generated in massive ST theory as compared with its massless counterpart: GW amplitudes can be orders of magnitude larger and the waves are spread out into a nearly monochromatic signal over years or even centuries due to the dispersion of the massive field. GW searches may therefore be directed at historic supernovae such as SN1987A and either observe a signal or constrain the theory's parameter space [1220].

Numerical studies of binary systems in ST theory are rather scarce. The no-hair theorems strongly constrain possible deviations of pure BH spacetimes from GR. They can be bypassed, however, through non-trivial potentials [1221] or boundary conditions [1103] which leads to scalar wave emission. Nevertheless, NS systems appear to be the more natural candidate to search for imprints of ST theory. Dynamical scalarization has indeed been observed in simulations of the merger of two NSs with initially vanishing scalar charge [1222, 1223]. Beyond GR and ST theory, we are only aware of one numerical study [1104], which simulated the evolution of BH binaries in the dynamical Chern-Simons (dCS) theory linearized around GR. LIGO observations may then measure or constrain the dCS length scale to $\lesssim$$\mathcal{O}(10)~{\rm km}$.

With the dawn of GW astronomy [1], the topics discussed so far in this section are becoming important subjects of observational studies with LIGO, Virgo and future GW detectors. These studies are still in an early stage and the generation of precision waveforms for scenarios involving modifications of gravity, fundamental fields or more exotic compact objects will be a key requirement for fully exploiting the scientific potential of this new channel to observing the universe.

The analysis of GW events has so far concentrated on testing the consistency of the observed signals with GR predictions, establishing bounds on phenomenological parametrizations of deviations from GR and obtaining constraints from the propagation of the GW signal. An extended study of GW150914 demonstrated consistency between the merger remnant's mass and spins obtained separately from the low-frequency inspiral and the high-frequency postinspiral signal [17]. The Compton wavelength of the graviton was constrained with a 90% lower bound of 10¹³ km (corresponding to an upper bound for the mass of  ∼$10^{-22}\,{\rm eV}$) and parametrizations of violations of GR using high-order PN terms have been constrained; see also [11, 20]. The first NSB observation [21], combined with electromagnetic observations, limited the difference between the speed of propagation of GWs and that of light to within $-3\times 10^{-15}$ and $7\times 10^{-16}$ times the speed of light [23]. Analysis of the polarization of the first triple coincidence detection GW170814 found Bayes' factors of 200 (1000) favoring a purely tensor polarization against purely vector (purely scalar) radiation. In summary, the GW observations have as yet not identified an inconsistency with the predictions of vacuum GR for BBH signals or GR predictions for NS systems.

Chapter III contains detailed discussions on many of the topics raised here, including motivation and brief description of certain broad classes of alternative to GR (section 2.1), numerics beyond GR (section 3.4), and the nature of compact objects beyond GR (section 4).

The hierarchy problem of physics consists in the vast discrepancy between the weak coupling scale ($\approx$246 ${\rm GeV}$) and the Planck scale $1.31\times 10^{19}~{\rm GeV}$ or, equivalently, the relative weakness of gravity compared with the other interactions. A possible explanation has been suggested in the form of 'large' extra spatial dimensions [1224, 1225] or extra dimensions with a warp factor [1226, 1227]. On short lengthscales $\lesssim$$10^{-4}~{\rm m}$, gravity is at present poorly constrained by experiment and would, according to these models, be diluted due to the steeper fall off in higher dimensions. All other interactions, on the other hand, would be constrained to a 3+1 dimensional brane and, hence, be unaffected. In these braneworld scenarios, the fundamental Planck scale would be much smaller than the four-dimensional value quoted above, possibly as low as $\mathcal{O}({\rm TeV})$ which inspired the name TeV gravity. This fundamental Planck scale determines the energy regime where gravity starts dominating the interaction, leading to the exciting possibility that BHs may be formed in particle collisions at the LHC or in high-energy cosmic rays hitting the atmosphere [1228–1230].

The analysis of experimental data employs so-called Monte-Carlo event generators [1231] which require as input the cross section for BH formation and the loss of energy and momentum through GW emission. In the ultrarelativistic limit, the particles may be modeled as pointlike or, in GR, as BHs.

In D  =  4 spacetime dimensions, high-energy collisions of BHs are by now well understood. Head-on collisions near the speed of light radiate about $14\%$ of the center-of-mass energy M in GWs [1232, 1233]. The impact parameter separating merging from scattering collisions with boost velocity $v$ is $b/M=(2.50\pm0.05)/v$ [1234]. Grazing collisions exhibit zoom-whirl behaviour [1235, 1236] and can radiate up to  ∼$50\%$ of the total energy in GWs [1237]. The collision dynamics furthermore become insensitive to the structure of the colliding objects—BHs or matter balls—at high velocities [1190, 1237–1240] as expected when kinetic energy dominates the budget. Finally, NR simulations of hyperbolic BH encounters are in good agreement with predictions by the EOB method [1022].

The key challenges are to generalize these results to the higher D scenarios relevant for TeV gravity (there are partial, perturbative results [1241–1243]). Considering the symmetries of the collision experiments, it appears plausible to employ rotational symmetry in higher D numerics which is vital to keep the computational costs under control and underlies all NR studies performed so far [1054, 1244–1247]. Nonetheless, NR in higher D faces new challenges. The extraction of GWs is more complex, but now tractable either with perturbative methods [1248–1250], using the Landau-Lifshitz pseudotensor [1251] or projections of the Weyl tensor [1252, 1253]. Likewise, initial data can be obtained by generalizing four-dimensional techniques [1254]. Studies performed so far, however, indicate that, for reasons not yet fully understood, obtaining numerically stable evolutions is harder than in D  =  4 [1255, 1256]. Results for the scattering threshold are limited to $v\lesssim 0.5~c$ [1255] and the emission of GWs has only been computed for non-boosted collisions of equal and unequal mass BH binaries [1256–1258]. These simulations show a strong suppression of the radiated energy with D beyond its peak value $E_{\rm rad}/M \approx 9\times 10^{-4}$ at D  =  5 for equal-mass systems, but reveal more complex behaviour for low mass ratios. A remarkable outcome of BH grazing collisions in D  =  5 is the possibility of super-Planckian curvature in a region outside the BH horizons [1255].

Recent years have seen a surge of NR applications to non-asymptotically flat spacetimes in the context of the gauge-gravity duality, cosmological settings and for the exploration of fundamental properties of BH spacetimes. We list here a brief selection of some results and open questions; more details can be found in the reviews [1259, 1260].

Cosmic censorship has for a long time been a topic of interest in NR, but to date no generic, regular and asymptotically flat class of initial data are known to result in the formation of naked singularities in four spacetime dimensions. Higher-dimensional BHs, however, have a much richer phenomenology [1261], including in particular black rings which may be subject to the Gregory–Laflamme instability [1262]. Thin black rings have indeed been found to cascade to a chain of nearly circular BHs connected by ever thinner segments in finite time [1263] in the same way as infinite black strings [1264]. Similarly, ultraspinning topologically spherical BHs in $D\geqslant 6$ dimensions are unstable [1265] and ultimately form ever thinner rings in violation of the weak cosmic censorship conjecture [1266].

We have already mentioned Choptuik's discovery of critical phenomena in the collapse of spherical scalar fields [1141]. In asymptotically anti-de Sitter (AdS) spacetimes, the dynamics change through the confining mechanism of the AdS boundary, allowing the scalar field to recollapse again and again until a BH forms [1267–1271]; see also [1272] for non-spherically symmetric configurations. NR simulations of asymptotically AdS spacetimes are very challenging due to the complex outer boundary conditions, in particular away from spherical symmetry, but recent years have seen the first simulations of BH collisions in AdS [1273, 1274] which, assuming gauge-gravity duality, would imply a far-from hydrodynamic behaviour in heavy-ion collisions during the early time after merger.

Using a cosmological constant with an opposite sign leads to asymptotically de Sitter (dS) spacetimes widely believed to describe the Universe we live in. NR studies in dS have explored the possible impact of local structures on the cosmological expansion [1275–1279] and found such inhomogeneities to not significantly affect the global expansion. Further work has explored the robustness of inflation under inhomogeneities [1280], the propagation of light in an expanding universe [1281] and the impact of extreme values of the cosmological constant on the physics of BH collisions [1282].

The EOB approach was introduced in [1286, 1287] as a method to combine information from the test-particle limit with PN results (for early ideas in this spirit, see [1288]). The model comprises a Hamiltonian for the inspiral dynamics, a prescription for computing the GWs and corresponding radiation reaction forces, and a smooth transition to the ringdown signals from a perturbed final BH. The idea is to map the conservative dynamics of the relative motion of two bodies, with masses m_1,2, spins ${\boldsymbol S}_{1,2}$, orbital separation $\boldsymbol x$ and relative momentum $\boldsymbol p$, onto an auxiliary Hamiltonian description of an effective particle of mass $\mu =m_1\,m_2/(m_1+m_2)$ and effective spin $\boldsymbol{S}_*(\boldsymbol{S}_1,\boldsymbol{S}_2,\boldsymbol{x},\boldsymbol{p})$ moving in an effective spacetime $g_{\mu \nu}^{\rm ef\,\!f}(M,\boldsymbol{S}_{\rm Kerr};\nu)$ that is characterized by the total mass $M =m_1+m_2$, symmetric mass ratio $\nu= \mu/M$, and total spin $\boldsymbol{S}_{\rm Kerr}(\boldsymbol{S}_1,\boldsymbol{S}_2)$. The basic requirements on this mapping are that (i) the test-particle limit reduces to a particle in Kerr spacetime, and (ii) in the weak-field, slow-motion limit the EOB Hamiltonian reduces to the PN Hamiltonian, up to canonical transformations. These considerations, together with insights from results in quantum-electrodynamics [1289], scattering theory [1004, 1032], and explicit comparisons of the action variables [1286] all lead to the 'energy map' given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H=M \sqrt{1+2\nu \left(\frac{H_{\rm ef\,\!f}}{\mu}-1\right)}, \nonumber \end{align} \tag{ 11 }$$

where H and $H_{\rm ef\,\!f}$ are the Hamiltonians describing the physical binary system and the effective particle respectively.

The details of the effective Hamiltonian $H_{\rm ef\,\!f}$ are less constrained by theoretical considerations, and different descriptions have been proposed that differ in the structure of the Hamiltonian, the form of the potentials characterizing the effective spacetime, and the spin mapping between the physical and effective systems. The differences between these choices reflect current limitations of theoretical knowledge; they all agree in the PN and nonspinning test-particle limit. For the structure of $H_{\rm ef\,\!f}$, the incarnation of the EOB model of [1108, 1290–1293] imposes that the limiting case of a spinning test-particle in Kerr spacetime must be recovered, to linear order in the test-spin. The version of the model of [1294–1298] does not include test-particle spin effects, which enables a more compact description. These different choices also result in different spin mappings, for both the Kerr parameter and the spin of the effective particle. Finally, these two branches of EOB models also employ different ways to re-write the potentials that are calculated as a Taylor series in a PN expansion in a 're-summed' form. This means that an empirically motivated non-analytic representation is used that consists either of a Pade-resummation [1294–1296] or has a logarithmic form [1291, 1292]. In addition to the above choices for describing the strong-field, comparable-mass regime, the models also include parameterized terms whose coefficients are functions of the mass and spin parameters that are fixed by comparisons to NR results. In the model of [1108, 1291–1293], these calibration parameters are constrained by the requirement that the model must reproduce the GSF results for the ISCO shift in Schwarzschild [813].

The radiative sector in the EOB model is described by so-called factorized waveforms (instead of a PN Taylor series expansion) that are motivated from the structure of waveforms in the test-particle limit and have the form [1299, 1300]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{hlm} h_{\ell m}^{\rm insp\mbox{-}plunge}(t)=h_{\ell m}^{(N,\epsilon)}\,\hat{S}_{\rm ef\,\!f}^{(\epsilon)}\, T_{\ell m}\, {\rm e}^{{\rm i}\delta_{\ell m}}\, f_{\ell m}\,N_{\ell m}. \nonumber \end{align} \tag{ 12 }$$

Here, $\newcommand{\e}{{\rm e}} h_{\ell m}^{(N,\epsilon)}$ is the Newtonian contribution, and $\newcommand{\e}{{\rm e}} \hat{S}_{\rm ef\,\!f}^{(\epsilon)}$ is a certain effective 'source term' that, depending on the parity $\newcommand{\e}{{\rm e}} \epsilon$ of the mode, is related to either the energy or the angular momentum. The factor $\newcommand{\e}{{\rm e}} T_{\ell m}$ contains the leading order logarithms arising from tail effects, the term $\newcommand{\e}{{\rm e}} {\rm e}^{{\rm i}\delta_{\ell m}}$ is a phase correction due to sub-leading order logarithms in hereditary contributions, while the function $\newcommand{\e}{{\rm e}} f_{\ell m}$ collects the remaining PN terms. Finally, the factor $\newcommand{\e}{{\rm e}} N_{\ell m}$ is a phenomenological correction that accounts for deviations from quasi-circular motion [1301] and is calibrated to NR results. The modes from equation (12) are used to construct dissipative forces $\boldsymbol{\mathcal{F}}$, in terms of which the equations of motion are given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{\rm d}\boldsymbol{x}}{{\rm d} t}=\frac{\partial H}{\partial \boldsymbol{p}}, \quad \quad \frac{{\rm d}\boldsymbol{p}}{{\rm d} t}=-\frac{\partial H}{\partial \boldsymbol{x}} +\boldsymbol{\mathcal{F}}, \qquad \qquad \nonumber \end{align} \tag{ 13 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{\rm d}\boldsymbol{S}_{1,2}}{{\rm d} t}=\frac{\partial H}{\partial \boldsymbol{S}_{1,2}}\times \boldsymbol{S}_{1,2}. \nonumber \end{align} \tag{ 14 }$$

This EOB description of the inspiral-plunge dynamics is smoothly connected to the merger-ringdown signal in the vicinity of the peak in the amplitude |h₂₂|. To perform this matching, initial nonspinning models and the aligned-spin models SEOBNRv1 [1292] and SEOBNRv2 [1293, 1302] (as well as in the models of the IHES group [1294, 1303]) used a superposition of damped sinusoids similar to quasi-normal modes [1287]. This method is inspired by results for the infall of a test-particle into a BH and the close-limit approximation [1304]. More recently, a simpler phenomenological fit of the amplitude and phase inspired by the rotating source approximation [1305], which was adapted to the EOB context in [1306], has become standard in SEOBNRv4 [1108]. This method provides a more stable and controlled way to connect the inspiral-plunge to the ringdown. A further key input into the merger-ringdown model is the frequency of the least-damped quadrupolar quasinormal mode $\sigma_{220}$ of the remnant based on a fitting formula from NR for the mass and spin of the final object given the initial parameters, with the currently most up-to-date fit from [1108].

Generic spin precession effects are also included in the model, by starting from a calibrated spin-aligned model and transforming to the precessing frame as dictated by the precession equations derived from the EOB Hamiltonian [1307, 1308], without further calibrations. The most recent refinement for generic precessing binaries from [1308] is known as SEOBNRv3.

For non-vacuum binaries involving e.g. neutron stars, exotic compact objects, or condensates of fundamental fields around BHs, several effects of matter change the GWs from the inspiral relative to those from a BH binary, as described in chapter III, section 2.4, where also the signatures from such systems generated during the merger and ringdown are covered. Effects during the inspiral include spin-induced deformations of the objects, tidal effects, the presence of a surface instead of an event horizon, tidal excitation of internal oscillation modes of the objects, and more complex spin-tidal couplings. Two classes of EOB models for such systems are currently available, corresponding to the two different baseline EOB models for BHs. One is known as TEOBRESUMS [1309] and incorporates the effects of rotational deformation and adiabatic tidal effects [1310, 1311] in re-summed form that was inspired by [826] and augmented as described in [439]. The other model is known as SEOBNRv4T and includes the spin-induced quadrupole as well as dynamical tides from the objects' fundamental oscillation modes [447, 1312].

The aim of the Phenom models is to provide a simple, efficient, closed-form expression for the GW signal in the frequency domain [1313] by assuming the schematic form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tilde{h}_{\rm phen}(\,f;\vec {\alpha}; \vec {\beta}) := A(\,f;\vec {\alpha}){\rm e}^{{\rm i}\Psi(\,f;\vec {\beta})}, \nonumber \end{align} \tag{ 15 }$$

where $\vec {\alpha}$ and $\vec {\beta}$ are amplitude and phase parameters in the model. Phenomenological ('Phenom') models were first developed for nonspinning binaries in [1313, 1314] and subsequently refined to include aligned spins [1315] known as 'PhenomB'. This model employed only a single weighted combination of the individual BH spins characterizing the dominant spin effect in the GWs, and was further refined in [1316] ('PhenomC'), and in [1097, 1107] ('PhenomD'). The latter has been calibrated using NR data for mass ratios up to 1:18 and dimensionless spins up to 0.85 (with a larger spin range for equal masses). An effective description of the dominant precession effects has also been developed [1111] ('PhenomP') [1317, 1318]. The PhenomP model provides an approximate mapping for obtaining a precessing waveform from any non-precessing model, with PhenomD being currently used as the baseline.

To construct these state-of-the-art Phenom models, the GW signal is divided into three main regimes: an inspiral, intermediate region, and merger-ringdown. The ansatz for the inspiral model is the PN result for the frequency-domain phasing obtained from energy balance in the stationary phase approximation ('TaylorF2'), accurate to 3.5PN in the nonspinning and linear-in-spin sector, and to 2PN in spin–spin terms. This has the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Psi_{\rm TF2}=2\pi f t_c-\phi_c-\pi/4+\frac{3}{128\nu}(\pi M f)^{-5/3}\sum_{i=0}^7 c_i (\pi M f)^{i/3}, \label{eq:TF2phase} \nonumber \end{align} \tag{ 16 }$$

where c_i are PN coefficients. The Phenom models add to this auxiliary terms so that the inspiral phase becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Psi=\Psi_{\rm TF2}+\frac{1}{\nu}\sum_{i=0}^6 \sigma_i f^{i/3}, \nonumber \end{align} \tag{ 17 }$$

where $\sigma_i$ are phenomenological coefficient with $\sigma_{1}=\sigma_2=0$.

A different ansatz is made for the late inspiral and merger-ringdown signals that are likewise closed-form expressions involving the frequency and phenomenological parameters; in total the model involves 17 phenomenological parameters. For the late inspiral portion, these are mapped to the set of two physical parameters $(\nu, \chi_{\rm PN})$, where $\chi_{\rm PN}$ is defined by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \chi_{\rm PN}= \chi_{\rm ef\,\!f}-\frac{38\nu}{113}\left(\chi_1+\chi_2\right), \nonumber \end{align} \tag{ 18 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \chi_{\rm ef\,\!f}=\frac{m_1}{M} \vec {\chi}_1\cdot \hat{L}_N+\frac{m_2}{M} \vec {\chi}_2\cdot \hat{L}_N,\label{chi} \nonumber \end{align} \tag{ 19 }$$

where $\hat L_N$ is the direction of the Newtonian angular momentum and $\chi_i=S_i/m_i^2$. $\chi_{\rm PN}$ describes the dominant spin effect (the dependence on M is only a rescaling). The mapping is done by assuming a polynomial dependence $\sigma_i=\sum b_{ijk}\nu^{\,j} \chi_{\rm ef\,\!f}^k$ to quadratic order in $\nu$ and third order in $\chi_{\rm PN}$ so that the coefficients vary smoothly across the parameter space. Finally, the coefficients are calibrated to a large set of hybrid waveforms, which themselves are formed using the uncalibrated SEOBNRv2 model.

Spin effects in the Phenom models are described by several different combinations of parameters. In the aligned-spin baseline model, the description of the early inspiral depends on both spin parameters $\chi_1,\chi_2$ from PN. In the later inspiral regime, spin effects are described by the effective combination $\chi_{\rm PN}$ described above, while the merger-ringdown model is expressed in terms of the total spin. For generic spin orientations, an additional parameter $\chi_p$ that characterized the most readily measurable effects of precession is included in the model. The defnition of the precessional parameter $\chi_p$ is motivated by the observation that waveforms are simpler in the coprecessing frame that is aligned with the direction of the dominant radiation emission [1044]; it is given by [1318]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \chi_p=\frac{1}{B_1 m_1^2}{\rm max}(B_1 S_{1\perp},B_2 S_{2\perp}), \nonumber \end{align} \tag{ 20 }$$

where $B_1=2+3 m_2/(2 m_1)$ and $B_2=2+3 m_1/(2 m_2)$ assuming the convention $m_1>m_2$. Starting from an aligned-spin frequency-domain waveform model, an approximate precessing waveform is constructed by 'twisting up' the non-precessing waveform with the precessional motion based on a single-spin PN description for the precession angles [1111, 1317]. While there are some broad similarities of PhenomP with the way the precessing EOB model is constructed, the two approaches differ in several details, as explained in section IV of [1308].

For non-BH objects, the effects of rotational and tidal deformations are included in the phasing using either PN information [1319–1321] or, for the case of binary neutron stars, using a model calibrated to NR [1322, 1323].

Important advances that all EOB and Phenom models aim to address in the near-future are higher modes, parameter space coverage (especially in the mass ratio) and inclusion of all available theoretical knowledge, reduction of systematic errors, and going beyond circular orbits.

To date, most of the effort in calibrating the EOB and Phenom models has focused on the $(2,2)$ mode, although for the special case of nonspinning binaries an EOB multipolar approximant is available [1324]. An accurate model of higher modes is important for robust data analysis, especially for spinning binaries. Work is ongoing to address this within the EOB model [1325] and with studies that will inform the Phenom approach [1326]. The parameter space over which current models have been calibrated and tested is limited by available NR simulations of sufficient accuracy and length (see section 4), and systematic uncertainties remain a concern.

Extending the range in parameter space ties into the pressing issue that many available results from GSF calculations are not currently incorporated in these models. One of the obstacles is that GSF data for circular orbits only make sense above the 'light ring' radius and cannot directly inform the EOB model across and below that radius. This issue is a further subject of ongoing work. Efforts are also underway to include effects of eccentricity [1327–1329]. Effects such as the motion of the center-of-mass, BH radiation absorption, and radiation reaction for spins are also not yet included in current models.

For EOB models, efficiency for data analysis is a further concern, because computing EOB waveforms is relatively time-consuming. To overcome this issue, reduced-order models for EOB waveforms in the frequency domain have been developed for aligned-spin binaries [1108, 1330], with work being underway on the larger parameter space of precessing binaries. Also ongoing is work on theoretical aspects of spin effects in the EOB model, aimed at devising an improved description that is more robust and computationally efficient.

The effective models described above have been used for tests of GR and exotic objects, e.g. in [17]. One of the parameterized tests makes use of the general framework 'TIGER' [1331] that has been implemented for data analysis, where parameterized fractional deviations from the GR coefficients are included in the frequency-domain phase evolution from equation (16). In recent studies, e.g. [17, 18], this framework was used on top of the PhenomD inspiral model [1097, 1107] to obtain bounds on non-GR effects, where the tests also included parameterized deviations in the intermediate and merger-ringdown regime. Work is ongoing to implement tests of GR within the EOB approach [1332]. Finally, work is also underway to include in these models the option to test for exotic physics manifest in spin-quadrupole and tidal effects during the inspiral. Despite this recent progress on frameworks to test GR and the nature of BHs, substantial further work will be necessary to refine the level of sophistication and physical realism of such tests.

The LIGO-Virgo Collaboration (LVC) uses a variety of techniques to first identify and then characterise GW transients identified in their data. The majority of these make use of signal models in some way. We briefly describe these methods in the following.

7.1.1. Unmodelled searches.

7.1.2. Template based searches.

7.1.3. Parameter estimation.

The next LIGO-Virgo observing run, O3, is scheduled to start in early 2019, and is expected to have a factor of  ∼2 improvement in sensitivity over the O2 run that finished in August 2017. LIGO/Virgo are expected to complete their first science runs at design sensitivity in the early 2020s [14]. Based on the earlier science runs, several tens of BBH systems and a small number of NSB events [9, 21] are likely to be detected in O3. These events will be similar to sources previously identified and so the primary challenge will be computational—the LVC will need the capability to process multiple events simultaneously, which implies a need for accurate waveform models that are as fast to generate as possible. Further in the future there are plans for third generation ground based detectors, such as the Einstein telescope [33] and cosmic explorer [32], and a space-based GW detector, the Laser Interferometer Space Antenna (LISA) [64]. These new detectors will observe new types of source which will pose new modelling challenges.

7.2.1. Third-generation ground-based detectors.

7.2.2. The Laser Interferometer Space Antenna.

As described above, a significant obstacle to data analysis for future detectors is computational. In order to analyse a large number of potential signals and search large parameter spaces, we require models that capture all the main physics reliably but can be evaluated rapidly. We describe the current status and outstanding challenges here.

7.3.1. Reduced-order models.

7.3.2. Kludges.

A working axiom in the GW community is that Kerr BHs exhaust the collection of BH end states. There is strong evidence for this, based on the 'no-hair' theorem, which asserts that suitably regular stationary, analytic, non-degenerate, connected, asymptotically flat vacuum BHs belong to the Kerr family; see [1376] for precise definitions and a list of many contributors (see also sections 4.1 and 4.2 of chapter III for a discussion on no-hair theorems in the context of beyond-GR phenomenology). The analyticity assumption is highly undesirable, being rather curious: it implies that knowledge of a metric in a small space-time region determines the metric throughout the universe. The analyticity assumption was relaxed in [1377] for near-Kerr configurations, but the general case remains open. Next, the non-degeneracy assumption is essentially that of non-vanishing surface gravity: the maximally spinning Kerr BHs are degenerate in this sense. One often discards the extremal-Kerr case by declaring that it is unstable. This might well be the case, but it is perplexing that the spin range of observed SMBH candidates is clearly biased towards high spin, with a possible peak at extremality [261, 1378]. (Note, however, that these studies have an a priori assumption built-in, that the BHs are Kerr.) Finally, while the connectedness assumption is irrelevant when talking about an isolated BH, it is quite unsatisfactory to have it around: what mechanism could keep a many-BH vacuum configuration in a stationary state? All this makes it clear that removing the undesirable assumptions of analyticity, non-degeneracy, and connectedness in the uniqueness theorems should be a high priority to mathematically-minded relativists.

The current paradigm is that, after a merger, the final BH settles to a stationary one by emitting radiation which, at the initial stage, has a characteristic profile determined by quasi-normal modes. This characteristic radiation field provides a very useful tool for determining the final mass and angular momentum of the solution. (Note that the extraction of these parameters from the ringdown profile assumes the final state to be Kerr, taking us back to the issues raised in the previous paragraph...) All this seems to be well supported by numerics. But we are still far behind by way of mathematical proof. Work by Dyatlov [1379, 1380] has rigorously established that quasi-normal modes are part of an asymptotic expansion of the time-behaviour of linear waves on the Kerr-de Sitter background with a nonzero cosmological constant. The asymptotically flat Kerr case turns out to be much more difficult to tackle, and so far no rigorous mathematical statements are available for it in the literature. The whole problem becomes even more difficult when non-linearities are introduced, with no results available so far. The recently announced proofs of nonlinear stability of the Schwarzschild solution within specific families of initial data [1381, 1382] might serve as a starting point for further studies of this important problem.

It was mind-blowing, and still is, when LIGO detected two BHs of about 35 and 30 solar masses merging into one, of about 60 solar masses. But the question arises, what do these numbers mean? Assuming the end state to be Kerr, the last part of the statement is clear. But how can the pre-merger masses be determined, or even defined, given the non-stationarity of the BBH configuration? Each NR group uses its own method for assigning mass and spin parameters to the initial datasets used in their simulations, and it is not a priori clear how these numbers relate to each other. EOB models likewise employ their own definitions of mass and spin. Any differences are most likely irrelevant at the current level of detection accuracy. But one hopes that it will become observationally important at some stage, and is a fundamental issue in our formulation of the problem in any case.

To address the issue requires choosing benchmark definitions of quasi-local mass, momentum, and angular momentum that can be blindly calculated on numerical initial datasets, without knowing whether the data were obtained, e.g. by solving the spacelike constraints with an Ansatz, or by evolving some other Ansatz, or by matching spacelike and characteristic data. One can imagine a strategy whereby one first locates the apparent horizons within a unique preferred time-slicing (e.g. maximal, keeping in mind that apparent horizons are slicing-dependent), and then calculates the chosen quasi-local quantities for those. There exists a long list of candidates for quasi-local mass, with various degrees of computational difficulty, which could be used after extensive testing, all to be found in [1383]; alphabetically: Bartnik's, Brown-York's, Hawking's (directly readable from the area), Kijowski's, Penrose's, with the currently-most-sophisticated one due to Chen, Wang and Yau (see [1384] and references therein). The issue is not whether there is an optimal definition of quasi-local quantities, which is an interesting theoretical question on its own, unlikely to find a universally agreed answer, but whether there is a well-defined and computationally convenient one that can be used for scientific communication. Such a definition would need to have an unambiguous Newtonian limit, necessary for making contact with non-GR astrophysical observations.

To an outsider, NR looks like a heroic struggle with a quasi-impossible task, which, after years of inspired attempts, resulted in a maze of incredibly complicated codes that manage to perform the calculations related to the problem at hand. It is conceivable that there is no way to control the whole construction in a coherent way. However, some mathematically minded outsiders would like to get convinced that the numerical calculations are doing what they are supposed to do; compare [1385, 1386]. In other words, is it possible to show that, at least some, if not most, if not all of the current numerical approximations to Einstein equations would converge to a real solution of the problem at hand if the numerical accuracy could be increased without limit? Standard convergence tests, or checks that the constraints are preserved, are of course very important, but they cannot on their own settle the point. A more rigorous proof of convergence is desirable.

The most straightforward way to test new fundamental physics with GWs from BBH coalescences would be the following: select one's favourite scenario and model the system in full detail, extract a waveform (or better a template bank), and then look for the prediction in the data. The technical difficulties of doing so will become apparent below, where it will also be apparent that the required tools are at best incomplete and require further development. However, there is a clear non-technical drawback in this approach as well: it assumes that one knows what new fundamental physics to expect and how to model it to the required precision. One should contrast this with the fact that most quantum gravity candidates, for example, are not developed enough to give unique and precise predictions for the dynamics of a binary system. Moreover, ideally one would want to obtain the maximal amount of information from the data, instead of just looking for very specific predictions, in order not to miss unexpected new physics. Hence, the question one needs to ask is what is the optimal way to extract new physics from the data?

GW observations test gravity itself and the way matter interacts through gravity. Hence, at the theoretical level, they can test GR and the way GR couples to the Standard Model (SM) and its extensions. This suggests clearly that the new fundamental physics that can leave an imprint on GW observations can most likely be effectively modelled using an alternative theory of gravity. Recall that alternative theories of gravity generically contain extra degrees of freedom that can be nonminimally coupled to gravity.

The advantages of this approach are: (i) alternative theories of gravity can act as effective field theories for describing certain effects and phenomena (e.g. violations of Lorentz symmetry or parity) or be eventually linked to a specific more fundamental theory; (ii) they can provide a complete framework for obtaining predictions for binary evolutions and waveforms, as they come with fully nonlinear field equations; (iii) their range of validity is broad, so they allow one to combine constraints coming from the strong gravity regime with many other bounds from e.g. the weak field regime, cosmology, astrophysics, laboratory tests, etc. The major drawback of this approach is that it requires theory-dependent modelling, which can be tedious and requires one to focus on specific alternative theories of gravity.

In this section we will give a brief overview of some alternative theories that can be used to model new physics in GW observations. This is not meant to be a comprehensive list nor is it our intent to pinpoint interesting candidates. We simply focus on theories that have received significant attention in the literature in what regards their properties in the strong gravity regime and to which we plan to refer to in the coming sections.

It is worth mentioning that a complementary approach is to use theory-agnostic strong field parametrisations. Their clear advantage is that they simplify the modelling drastically and they render it theory-independent. However, they fall short in points (i)–(ii) above. In specific, it is no always straightforward to physical interpret them, they typically describe only part of the waveform, and they do not allow one to combine constraints with other observations unless interpreted in the framework of a theory. Strong field parametrisations and their advantanges and disadvantages will be discussed in sections 3.1–3.3.

Contributors: T P Sotiriou, K Yagi

One of the simplest ways to modify GR is to introduce a scalar field that is nonminimally coupled to gravity. In many cases, the term scalar-tensor theories refers to theories described by the action

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{staction} S_{\rm st}=\frac{1}{16\pi }\int {\rm d}^4x \sqrt{-g} \Big(\varphi R-\frac{\omega(\varphi)}{\varphi} \nabla^\mu \varphi \nabla_\mu \varphi-V(\varphi) \Big) +S_m(g_{\mu\nu},\psi), \nonumber \end{align} \tag{ 21 }$$

where g is the determinant of the spacetime metric $g_{\mu\nu}$, R is the Ricci scalar of the metric, and S_m is the matter action. $\psi$ is used to collectively denote the matter fields, which are coupled minimally to $g_{\mu\nu}$. The functions $\omega(\varphi)$ and $V(\varphi)$ need to be specified to identify a specific theory within the class. Brans–Dicke theory is a special case with $\omega=\omega_0=$constant and $V=0$ [1195].

It is common in the literature to rewrite the action in terms of a different set of variables. In particular, the conformal transformation $\hat{g}_{\mu\nu}=G \varphi \, g_{\mu\nu}$, where G is Newton's constant, and the scalar field redefinition $2\varphi {\rm d}\phi=\sqrt{2\omega(\varphi)+3} \, {\rm d}\varphi$, can be employed to bring the action (21) to the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{stactionein} S_{\rm st}=\frac{1}{16\pi G}\int {\rm d}^4x \sqrt{-\hat{g}} \Big(\hat{R}-2 \hat{g}^{\nu\mu}\partial_\nu \phi \partial_\mu \phi-U(\phi)\Big)+S_m(g_{\mu\nu},\psi), \nonumber \end{align} \tag{ 22 }$$

where $U(\phi)=V(\varphi)/(G\varphi){}^2$. The set of variables $(\hat{g}_{\mu\nu}, \phi)$ is known as the Einstein frame, whereas the original set of variables $(g_{\mu\nu},\varphi)$ is known as the Jordan frame. Quantities with a hat are defined with $\hat{g}_{\mu\nu}$. In the Einstein frame, scalar-tensor theories seemingly take the form of Einstein's theory with a minimally coupled scalar field (hence the name). However, this comes at a price: $\phi$ now couples to the matter field $\psi$, as can be seen if S_m is written in terms of $\hat{g}_{\mu\nu}$, $\phi$, and $\psi$. Hence, if one sticks to the Einstein frame variables, one would infer that matter experiences an interaction with $\phi$ (fifth force), and hence it cannot follow geodesics of $\hat{g}_{\mu\nu}$. In the Jordan frame instead, there is no interaction between $\varphi$ and $\psi$ (by definition), and this implies that matter follows geodesics of $g_{\mu\nu}$. This line of thinking allows one to ascribe a specific physical meaning to the Jordan frame metric, but it does not change the fact that the two frames are equivalent descriptions of the same theory (see [1387] and references therein for a more detailed discussion).

Scalar-tensor theories are well-studied but generically disfavoured by weak field observations, e.g. [1203, 1388]. In brief, weak field tests dictate that the scalar field, if it exists, should be in a trivial configuration around the Sun and around weakly gravitating objects, such as test masses in laboratory experiments. These precision tests translate to very strong constraints on scalar-tensor theories; strong enough to make it unlikely that there can be any interesting phenomenology in the strong gravity regime, which is our focus here. There is one notable known exception [1201]. To introduce these models it is best to first set $V=U=0$ and then consider the equation of motion for $\phi$, as obtained from varying action (22). It reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \hat{\Box} \phi=-4\pi G \alpha(\phi) T , \nonumber \end{align} \tag{ 23 }$$

where $\newcommand{\e}{{\rm e}} T\equiv T^\mu_{\phantom{a}\mu}$, $\newcommand{\e}{{\rm e}} T_{\mu\nu}\equiv -2 (-g){}^{-1/2} \delta S_m/\delta g^{\mu\nu}$ is the Einstein frame stress energy tensor, and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \alpha(\phi)=- [2\omega(\varphi)+3]^{-1/2}. \nonumber \end{align} \tag{ 24 }$$

Solution with $\phi=\phi_0=$ constant are admissible only if $\alpha(\phi_0)=0$. This condition identifies a special class of theories and translates to $\omega(\varphi_0)\to \infty$ in the Jordan frame, where $\varphi_0$ is the corresponding value of $\varphi$. For $\phi=\phi_0$ solutions the corresponding metric is identical to the GR solution for the same system. However, these solutions are not unique. It turns out that they are preferable for objects that are less compact than a certain threshold, controlled by the value of $\alpha'(\phi_0)$. When compactness exceeds the threshold, a nontrivial scalar configuration is preferable, leading to deviations from GR [1197, 1201, 1389]. The phenomenon is called spontaneous scalarization and it is known to happen for neutron stars (NSs).

A NSB endowed with non-trivial scalar monopolar configurations would lose energy through dipole emission [1390] and this would affect the orbital dynamics. More specifically, this would be an additional, lower-order, contribution to the usual quadrupolar GW emission that changes the period of binary pulsars. Hence, binary pulsar constraints are extremely efficient in constraining spontaneous scalarization and the original scenario is almost ruled out [1391, 1392], with the exception of rapidly rotating stars. However, dipolar emission can be avoided if the scalar-scalar interaction between the stars is suppressed. This is the case is the scalar is massive [1209, 1219]. The mass defines a characteristic distance, beyond which the fall-off of the scalar profile is exponential (as opposed to 1/r), so if the value of the mass is in the right range, the stars can be scalarized and yet the binary will not exhibit any appreciable dipolar emission above a given separation. One can then avoid binary pulsar constraints and still hope to see some effect in GW emission from the late inspiral, merger, and ringdown. There is also the possibility that NSs will dynamically scalarize once they come close enough to each other [1222, 1223, 1393–1395].

It should be noted that the known scalarization scenario faces the following issue. Usually, calculations assume flat asymptotics with a vanishing gradient for the scalar. However, realistically the value of the scalar far away from the star changes at cosmological timescales and it turns out that this change is sufficient to create conflict with observations [1396, 1397]. This problem can be pushed into the future by tuning initial data, so it can technically be easily avoided. Nonetheless, one would eventually need to do away with the tuning, presumably by improving the model.

Black holes in scalar-tensor theories have received considerable attention, most notably in the context of no-hair theorems (e.g. [1398–1400]) and their evasions (see also [1401] for a discussion). Sections 4.1 and 4.2 are entirely devoted to this topic, so we will not discuss it further here.

The action of equation (21) is the most general one can write that is quadratic in the first derivatives of the scalar, and hence it presents itself as a sensible effective field theory for a scalar field nonminimally coupled to the metric. However, this action can be generalized further if one is willing to include more derivatives. Imposing the requirement that variation leads to field equations that are second order in derivatives of both the metric and the scalar leads to the action

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{hdaction} S_H=\int {\rm d}^4 x \sqrt{-g}\left(L_2+L_3+L_4+L_5\right), \nonumber \end{align} \tag{ 25 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_2 = K(\phi,X) , \nonumber \end{align} \tag{ 26 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_3 = -G_3(\phi,X) \Box \phi , \nonumber \end{align} \tag{ 27 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_4 = G_4(\phi,X) R + G_{4X} \left[ (\Box \phi)^2 -(\nabla_\mu\nabla_\nu\phi)^2 \right] , \nonumber \end{align} \tag{ 28 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_5 &= G_5(\phi,X) G_{\mu\nu}\nabla^\mu \nabla^\nu \phi \nonumber \\ &\qquad - \frac{G_{5X}}{6} \left[ (\Box \phi)^3 - 3\Box \phi(\nabla_\mu\nabla_\nu\phi)^2 + 2(\nabla_\mu\nabla_\nu\phi)^3 \right] , \nonumber \end{align} \tag{ 29 }$$

where $\newcommand{\e}{{\rm e}} X\equiv -\frac{1}{2} \nabla^\mu \phi \nabla_\mu \phi$, G_i need to be specified to pin down a model within the class, and $\newcommand{\e}{{\rm e}} G_{iX}\equiv \partial G_i/\partial X$. Horndeski was the first to write down this action in an equivalent form [1402] and it was rediscovered fairly recently in [1403]. Theories described by this action are referred to as generalized scalar-tensor theories, Horndeski theories, generalized galileons, or simply scalar-tensor theories. We will reserve the last term here for the action (21) in order to avoid confusion. The term generalized galileons comes from the fact that these theories were (re)discovered as curved space generalization of a class of flat space scalar theories that are symmetric under $\phi \to \phi + c_\mu x^\mu +c$, where $c_\mu$ is a constant one-form and c is a constant (Galilean symmetry) [1404]. In fact, the numbering of the L_i terms in the Lagrangian is a remnant of the original Galileons, where the i index indicates the number of copies of the field. This is no longer true in action (25), but now the L_i term contains i  −  2 second derivatives of the scalar.

In what regards strong gravity phenomenology, a lot of attention has been given to the shift-symmetric version of action (25). If $G_i=G_i(X)$, i.e. $\partial G_i/\partial \phi=0$, then (25) is invariant under $\phi \to \phi+$constant. This symmetry protects the scalar from acquiring a mass from quantum corrections. It has been shown in [1405] that asymptotically flat, static, spherically symmetric BHs in shift-symmetric Horndeski theories have trivial (constant) scalar configuration and are hence described by the Schwarzschild solution. This proof can be extended to slowly-rotating BHs, in which case the solution is the slowly rotating limit of the Kerr spacetime [1406]. However, there is a specific, unique term in (25) that circumvents the no-hair theorem and leads to BHs that differ from those of GR and have nontrivial scalar configurations (hairy BHs) [1406]: $\phi {{\mathcal G}}$, where $\newcommand{\e}{{\rm e}} {{\mathcal G}}\equiv R^2 - 4 R_{\mu\nu} R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ is the the Gauss–Bonnet invariant¹³⁷. This singles out

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{phiG} S_{\phi{{\mathcal G}}}=\frac{1}{16\pi G}\int {\rm d}^4x \sqrt{-g} \Big(R-\partial^\nu \phi \partial_\mu \phi+\alpha\phi {{\mathcal G}}\Big)+S_m(g_{\mu\nu},\psi), \nonumber \end{align} \tag{ 30 }$$

as the simplest action within the shift-symmetric Horndeski class that has hairy BHs. $\alpha$ is a coupling constant with dimension of a length squared. Hairy BHs in this theory [1408, 1409] are very similar to those found earlier in the (non-shift-symmetric) theory with exponential coupling [1410, 1411], i.e. $e^\phi {{\mathcal G}}$. The theory with exponential coupling is known as Einstein-dilaton Gauss–Bonnet theory and it arises as a low-energy effective theory of heterotic strings [1412, 1413].

The subclass of theories described by the action

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{fG} S_{f(\phi){{\mathcal G}}}=\frac{1}{16\pi G}\int {\rm d}^4x \sqrt{-g} \Big(R-\frac{1}{2}\partial^\nu \phi \partial_\mu \phi+f(\phi){{\mathcal G}}\Big)+S_m(g_{\mu\nu},\psi), \nonumber \end{align} \tag{ 31 }$$

seems to have interesting BH phenomenology in general. This theory admits GR solutions if $f'(\phi_0)=0$ for some constant $\phi_0$. Assuming this is the case, it has recently been proven in [1414] that stationary, asymptotically flat BH solutions will be identical to those of GR, provided $f''(\phi_0){{\mathcal G}}<0$. A similar proof, but restricted to spherical symmetry, can be found in [1415]. Theories that do not satisfy the $f''(\phi_0){{\mathcal G}}<0$ exhibit an interesting phenomenon [1414, 1416]: BH scalarization, similar to the NS scalarization discussed above. See also [1417] for a study of hairy BH solution in this class of theories and [1418] for a very recent exploration of linear stability.

The same class of theories can exhibit spontaneous scalarization in NSs [1414]. On the other hand, in shift-symmetric Horndeski theories it is known that for NSs the scalar configuration will be trivial [1419–1421], provided that the $\phi {{\mathcal G}}$ terms is absent. Even if it is there, there will be a faster than 1/r fall-off [1030, 1422]. Hence, for shift-symmetric theories BH binaries are probably the prime strong gravity system of interest.

It should be noted that the speed of GWs in generalized scalar tensor theories can differ from the speed of light under certain circumstances and this has been used to obtain constraint recently [670–673, 1423, 1424]. These will be discussed in section 3.1.

As mentioned above, action (25) is the most general one that leads to second order equations upon variation. Nonetheless, it has also been shown that there exist theories outside this class that contain no other degrees of freedom than the metric and the scalar field [1425–1430]. These models are often referred to as beyond-Horndeski theories.

Last but not least, dynamical Chern–Simons (dCS) gravity [1431, 1432] is a scalar-tensor theory that introduces gravitational parity violation at the level of the field equations. It draws motivation from the standard model [1433], heterotic superstring theory [1434], loop quantum gravity [1435–1439] and effective field theories for inflation [1440]. dCS gravity is not a member of the Horndeski class. The scalar is actually a pseudo-scalar, i.e. it changes sign under a parity transformation. Moreover, it couples to the Pontryagin density $*R^{\alpha\phantom{a}\gamma\delta}_{\phantom{a}\beta} R_{\phantom{a}\alpha\gamma\delta}^{\beta}$, where $R_{\phantom{a}\beta\gamma\delta}^{\alpha}$ is the Riemann tensor, $\newcommand{\e}{{\rm e}} *R^{\alpha\phantom{a}\gamma\delta}_{\phantom{a}\beta} \equiv {1\over2} \epsilon^{\gamma\delta\mu\nu}R_{\phantom{a}\beta\mu\nu}^{\alpha}$ and $\newcommand{\e}{{\rm e}} \epsilon^{\gamma\delta\mu\nu}$ is the Levi-Civita tensor. This coupling term gives rise to third-order derivatives in the field equations. The fact that the theory has higher-order equations implies that it is unlikely it is actually predictive as it stands [1106] (see also [1441]). Hence, most work has used a specific approach to circumvent that problem that is inspired by effective field theory and relies on allowing the coupling term to introduce only perturbative correction. This issue will be discussed in some detail in section 3.4.

dCS gravity has received a lot of attention in the strong gravity regime. Non-rotating BH and NS solutions are the same as in GR, as such solutions do not break parity. On the other hand, slowly-rotating [1442–1444], rapidly-rotating [1445] and extremal [1446, 1447] BHs and slowly-rotating NSs [1031] all differ from the corresponding GR solutions. In particular, such rotating compact objects acquire a scalar dipole hair and a correction to the quadrupole moment, which modify gravitational waveforms from compact binary inspirals [1448–1451] and mergers [1104]. Future GW observations will allow us to place bounds on the theory that are orders of magnitude stronger than the current bounds from Solar System [1452] and table-top experiments [1444]. Another interesting phenomenon in dCS gravity is gravitational amplitude birefringence, where the right-handed (left-handed) circularly-polarized GWs are either enhanced or suppressed (suppressed or enhanced) during their propagation from a source to Earth, which can be used to probe the evolution of the pseudo-scalar field [1453–1455]. Such GW observations are complementary [1456] to Solar System [1457] and binary pulsar [1458, 1459] probes.

For a general introduction to standard scalar-tensor theories see [1194, 1460] and for generalized scalar-tensor theories see [1461], and for a recent brief review on gravity and scalar fields see [1462].

Contributor: T P Sotiriou

Lorentz symmetry is a fundamental symmetry for the standard model of particle physics, hence it is important to question whether it plays an equally important role in gravity. To address this question one needs to study Lorentz-violating (LV) theories of gravity for two distinct reasons: (i) to quantify how much one is allowed to deviated from GR without contradicting combined observations that are compatible with Lorentz symmetry one needs to model the deviations in the framework of a consistent theory; (ii) studying the properties of such theories can provide theoretical insights. We will briefly review two examples of LV theories here.

Einstein-æther theory (æ-theory) [1463] is described by the action

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Saetheory} S_{ae}= \frac{1}{16\pi G}\int {\rm d}^{4}x\sqrt{-g} (-R -M^{\alpha\beta}{}_{\mu\nu} \nabla_\alpha u^\mu \nabla_\beta u^\nu) \nonumber \end{align} \tag{ 32 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle M^{\alpha\beta}{}_{\mu\nu} = c_1 g^{\alpha\beta}g_{\mu\nu}+c_2\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu} +c_3 \delta^{\alpha}_{\nu}\delta^{\beta}_{\mu}+c_4 u^\alpha u^\beta g_{\mu\nu}, \nonumber \end{align} \tag{ 33 }$$

c_i are dimensionless coupling constants and the æ ther $u_\mu$ is forced to satisfy the constraint $\newcommand{\e}{{\rm e}} u^2\equiv u^\mu u_\mu=1$. This constraint can be enforced by adding to the action the Lagrange multiplier term $\lambda (u^2-1)$ or by restricting the variation of the æ ther so as to respect the constraint. Due to the constraint, the æ ther has to be timelike, thereby defining a preferred threading of spacetime by timelike curves, exactly like an observer would. It cannot vanish in any configuration, including flat space. Hence, local Lorentz symmetry, and more precisely boost invariance, is violated. (32) is the most general action that is quadratic in the first derivatives of a vector field that couples to GR and satisfies the unit constraint. Indeed, æ -theory is a good effective field theory for Lorentz-symmetry breaking with this field content [1464]. It should be emphasised that the æ ther brings three extra degrees of freedom into the theory, two vector modes and a scalar (longitudinal) mode. The speed of any mode, including the usual spin-2 mode that corresponds to GWs, can differ from the speed of light and is controlled by a certain combination of the c_i parameters.

One can straightforwardly obtain an LV theory with a smaller field content from action (32) by restricting the æ ther to be hypersurface orthogonal [1465]. This amounts to imposing

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{ho} u_\mu=\frac{\partial_\mu T}{\sqrt{g^{\lambda\nu}\partial_\lambda T \partial_\nu T}}\, \nonumber \end{align} \tag{ 34 }$$

where T is a scalar and the condition is imposed before the variation (i.e. one varies with respect to T). This condition incorporates already the unit constraint u²  =  1 and removes the vector modes from the theory. It can be used to rewrite the action in terms of the metric and T only. T cannot vanish in any regular configuration that solves the field equations and its gradient will always be timelike. Hence, the æ ther is always orthogonal to some spacelike hypersurfaces over which T is constant. That is, T defines a preferred foliation and acts as a preferred time coordinate. Note that it appears in the action only through $u_\mu$ and the latter is invariant under $T\to f(T)$, so this is a symmetry of the theory and the preferred foliation can be relabelled freely.

One can actually use the freedom to choose the time coordinate and write this theory directly in the preferred foliation [1465]. Indeed, if T is selected as the time coordinate, the T  =  constant surfaces define a foliation by spacelike hypersurfaces with extrinsic curvature K_ij and N and Nⁱ are the lapse function and shift vector of this foliation. Then $u_\mu=N \delta^0_\mu$, and action (32) takes the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{SBPSHIR} S_T= \frac{1}{16\pi G'}\int {\rm d}T {\rm d}^3x \, N\sqrt{h}\left(K_{ij}K^{ij} - \lambda K^2 + \xi {}^{(3)}\!R + \eta a_ia^i \right), \nonumber \end{align} \tag{ 35 }$$

where $\newcommand{\e}{{\rm e}} a_i\equiv \partial_i \ln N$ and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{HLpar} \frac{G'}{G}=\xi=\frac{1}{1-(c_{1}+c_3)}, \quad \lambda=\frac{1+c_2}{1-(c_{1}+c_{3})},\quad \eta=\frac{c_1+c_4}{1-(c_1+c_3)}. \nonumber \end{align} \tag{ 36 }$$

Since T was chosen as a time coordinate, action (35) is no longer invariant under general diffeomorphisms. It is, however, invariant under the subset of diffeomorphisms that preserve the folation, or else the transformations $T\to T'=f(T)$ and $x^i\to x'^i=x'^i(T,x^i)$, which are now the defining symmetry of the theory from a EFT perspective. In fact, this action can be thought of as the infrared (low-energy) limit of a larger theory with the same symmetry, called Hořava gravity [1466]. The most general action of Hořava gravity is [1467]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{SBPSHfull} S_{H}= \frac{1}{16\pi G_{H}}\int {\rm d}T {\rm d}^3x \, N\sqrt{h}\left(L_2+\frac{1}{M_\star^2}L_4+\frac{1}{M_\star^4}L_6\right), \nonumber \end{align} \tag{ 37 }$$

where L₂ is precisely the Lagrangian of action (35). L₄ and L₆ collectively denote all the operators that are compatible with this symmetry and contain four and six spatial derivatives respectively (in the T foliation). The existence of higher order spatial derivatives implies that perturbation will satisfy higher order dispersion relations and $M_\star$ is a characteristic energy scale that suppresses the corresponding terms. Hořava gravity has been argued to be power-counting renormalizable and hence a quantum gravity candidate [1466]. The basic principle is that the higher-order spatial derivative operators modify the behaviour of the propagator in the ultraviolet and remove/regulate divergences (see [1468, 1469] for a discussion in the context of a simple scalar model). The projectable version of the theory [1466, 1470], in which the lapse N is restricted to have vanishing spatial derivatives, has actually been shown to be renormalizable (beyond power-counting) in four dimensions [1471], whereas the 3-dimensional version [1472] is asymptotically [1473]. This restricted version does however suffer from serious viability issues at low energies [1466, 1474–1477].

We will not discuss the precise form of L₄ and L₆ any further, as it not important for low-energy physics. It should be stressed though that such terms lead to higher order corrections to the dispersion relation of GWs and that of the extra scalar excitation Hořava gravity propagates. This implies that the speeds of GW polarizations can differ from the speed of light and acquire a frequency dependence. The relevant constraints from GW observations are discussed in section 3.1 (see also [1478] for a critical discussion). At large wavenumber k, the dispersion relations scale as $\omega^2 \propto k^6$ (since by construction the leading order ultraviolet operators in the action contain two temporal and six spatial derivatives) and this implies that an excitation can reach any speed, provided it has short enough wavelength. More surprisingly, it has been shown that Hořava gravity exhibits instantaneous propagation even in the infrared limit, described by action (35). In this limit, dispersion relations are linear but, both at perturbative [1479] and nonperturbative level [1480], there exists an extra degree of freedom that satisfies an elliptic equation on constant preferred time hypersurfaces. This equation is not preserved by time evolution and hence it transmits information instantaneously within these hypersurfaces. For a detailed discussion see [1481].

It should be clear from the discussion above that both æ -theory and Hořava gravity can exhibit superluminal propagation and this makes BHs particularly interesting in these theories. For both theories BHs differ from their GR counterparts and have very interesting feature that will be discussed further in section 4.2. For a review see [1482].

For early reviews on æ -theory and Hořava gravity see [1483] and [1484], respectively. For the most recent combined constraints see [1485, 1486] and references therein.

Contributor: S F Hassan

In the appropriate limit GR reproduces Newton's famous inverse square law of gravitation. At the same time, linear excitations in GR (GWs) satisfy a linear dispersion relation without a mass term. In particle theory terminology, both of these statements imply that the graviton—the boson that mediates the gravitational interaction—is massless. Extensions of GR that attempt to give the graviton a mass are commonly referred to as massive gravity, bimetric gravity, and multimetric gravity theories. One of the original motivations for the study of these theories was the hope that they could provide a resolution to the cosmological constant problem by changing the gravitational interaction beyond some length scale (controlled by the mass of the graviton). However, a more fundamental motivation emerges from comparing the structure of GR with that of the standard model of particle physics that successfully accounts for all known particles interactions. This is elaborated below.

The standard model describes particle physics in terms of fields of spin 0 (the Higgs field), spin 1/2 (leptons and quarks) and spin 1 (gauge bosons). At the most basic level, these fields are governed by the respective field equation, namely, the Klein–Gordon equation, the Dirac equation and the Maxwell/Yang–Mills/Proca equations. The structures of these equations are uniquely fixed by the spin and mass of the field along with basic symmetry and consistency requirements, in particular, the absence of ghost instabilities, for example, in spin-1 theories. From this point of view, GR is the unique ghost-free theory of a massless spin-2 field; the metric $g_{\mu\nu}$. Then the Einstein equation, $R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R=0$, with all its intricacies is simply the counterpart for a spin 2 field of the massless Klein–Gordon equation $\Box \phi=0$, or of the Maxwell equation $\partial_\mu F^{\mu\nu}=0$. However, to describe physics, the standard model contains a second level of structure: At the fundamental level, all fields appear in multiplets allowing them to transform under symmetries, in particular under gauge symmetries which are in turn responsible for the renormalizability and hence, for the quantum consistency of the theory. The observed fields are related to the fundamental ones through spontaneous symmetry breaking and through mixings. For example, to obtain a single Higgs field, one needs to start with four scalar fields arranged in a complex $SU(2)$ doublet, rather than the one scalar field of the final theory. Similarly, although electrodynamics is very well described by Maxwell's equations as a standalone theory, its origin in the standard model is much more intricate, requiring the spontaneous breaking of a larger gauge symmetry and a specific mixing of the original gauge fields into the vector potential of electromagnetism. Now, one may contrast these intricate features of the standard model with GR which is the simplest possible theory of a single massless spin 2 field with no room for any further structure. The comparison suggests embedding GR in setups with more than one spin 2 field and investigating the resulting models. The aim would be to see if the new structures could address some of the issues raised above, either in analogy with the standard model, or through new mechanisms. This will shed light on a corner of the theory space that has remained unexplored so far.

However, implementing this program is not as straightforward as it may seem. It was known since the early 70's that adding other spin 2 fields to GR generically leads to ghost instabilities. The instability was first noted in the context of massive gravity, which contains an interaction potential between the gravitational metric $g_{\mu\nu}$ and a nondynamical predetermined metric $f_{\mu\nu}$. However it also plagues dynamical theories of two or more spin 2 fields. Recent developments since 2010 have shown that one can indeed construct ghost free theories of massive gravity (with a fixed auxiliary metric) [1487, 1488], or fully dynamical ghost free bimetric theories [1489]. In particular, the bimetric theory is the theory of gravitational metric that interacts with an extra spin-2 field in specific ways and contains several parameters. It propagates one massless and one massive spin-2 mode which are combinations of the two metrics. Consequently, it contains massive gravity and GR as two different limits. For a review of bimetric theory see [1490].

In the massive gravity limit, as the name suggests, the theory has a single massive spin-2 field that mediates gravitational interactions in the background of a fixed auxiliary metric. The most relevant parameter then is the graviton mass which is constrained by different observations (for a review see, [1491]). In particular, consistency with the detection of GWs by LIGO restricts the graviton mass to $m < 7.7\times 10^{-23}~{\rm eV}$ [11, 17]. However, in this limit, it becomes difficult to describe both large scale and small scale phenomena consistently with the same fixed auxiliary metric.

Of course, the bimetric theory is more relevant phenomenologically close to the GR limit, where gravity is described by a mostly massless metric, interacting with a massive spin-2 field. Now the two most relevant parameters are the spin-2 mass m and the Planck mass M of the second spin-2 field, $M\ll M_P$, where M_P is the usual Planck mass of the gravitational metric. The presence of the second parameter M significantly relaxes the bounds on the spin-2 mass m. In the most commonly assumed scenarios, the stability of cosmological perturbations in the very early Universe requires $M \lesssim 100~ {\rm GeV}$ [1492]. In this case, GWs will contain a very small admixture of the massive mode, hence, m is not strongly constrained by LIGO and can be large. A new feature is that now the massive spin-2 field can be a DM candidate. Depending on the specific assumptions, the relevant allowed mass ranges are estimated as $10^{-4}~ {\rm eV}\lesssim m \lesssim 10^7~ {\rm eV}$ [1493], or the higher range, $1 ~{\rm TeV} \lesssim m \lesssim 100 ~{\rm TeV}$ [1494].

While some studies of cosmology in bimetric theory have been carried out, many important features of the theory are yet to be fully explored. One of these is the problem of causality and the well-posedness of the initial value problem. In particular, bigravity provides a framework for investigating models where the causal metric need not always correspond to the gravitational metric [1495, 1496]. Another interesting question is if the bimetric interaction potential can be generated through some kind of Higgs mechanism, in analogy with the Proca mass term for spin 1 fields. From a phenomenological point of view, it is interesting to investigate the massive spin 2 field present in the theory as a DM candidate. Strong field effects in bimetric gravity are expected to show larger deviations from GR as compared to weak field effects.

While ghost free bimetric models have interesting properties, they do not admit enlarged symmetries, since the two spin 2 fields have only nonderivative interactions. Constructing ghost free theories of spin 2 multiplets with gauge or global symmetries is a more difficult problem which requires constructing ghost free derivative interactions. So far, this is an unresolved problem which needs to be further investigated. If such theories exist, they will provide the closest analogues of the standard model in the spin 2 sector.

Contributor: T P Sotiriou

According to GR, GWs are massless and propagate at the speed of light, c. Hence, they satisfy the dispersion relation $E^2=c^2 p^2$, where E is energy and p  is momentum. Deviations from GR can introduce modifications to this dispersion relation. LIGO uses the following parametrisation to model such deviations: $E^2=c^2 p^2+A c^\alpha p^\alpha$, $\alpha>0$ [1498]¹³⁸. When $\alpha=0$, A represents a mass term in the dispersion relation. The current upper bound on the the mass from GW observations is $7.7\times 10^{-23}~{\rm eV}/c^2$ [11]. When $\alpha=2$, A represents a correction to the speed of GWs. The fact that GWs from the NSB merger GW170817 almost coincided with the gamma ray burst GRB 170817A has given a spectacular double-sided constraint on the speed of GWs, which should agree with the speed of light to less than a part in 10¹⁵ [21, 23].

All other corrections introduce dispersive effects which accumulate with the distance the wave had to travel to reach the detector. This enhances the constraints in principle. However, it should be noted that A has dimensions $[A]=[E]^{2-\alpha}$, which implies that corrections that have $\alpha>2$ are suppressed by positive powers of $c p/M_\star$. $M_\star$ denotes some characteristic energy scale and it is useful to think of the constraints on A as constraints on $M_\star$. GWs have long wavelengths, so one expects $\alpha>2$ corrections to be heavily suppressed unless the aforementioned energy scale is unexpectedly small. For instance, for $\alpha=3$ the current upper bound on $M_\star$ is in the $10^9~{\rm GeV}$ range, whereas for $\alpha=4$ it drops well below the eV range [11, 1478, 1498, 1500].

It is important to stress that all of the aforementioned constraints need to be interpreted within the framework of a theory (as is the case for any parametrisation). The mass bound can be turned to a constraint on the mass of the graviton in massive gravity theories, discussed in section 2.4. Interestingly, it is not the strongest constraint one can obtain [1491, 1501]. All other bounds can in principle be interpreted as constraints on Lorentz symmetry. However, terms with odd powers of $\alpha$ require spatial parity violations as well and the bounds on terms with $\alpha>2$ are probably too weak to give interesting constraints on Lorentz-violating gravity theories [1478]. The double-sided bound on the speed of GWs certainly yields a very strong constraint on Lorentz-violating theories of gravity. However, one has to take into account that such models generically have a multidimensional parameter space even at low energies and generically exhibit extra polarisations, the existence/absence of which can potentially give stronger or additional constraints [1478]. For details on the implications of this bound for Hořava gravity and for Einstein-æther theory, discussed in section 2.3, see [1485] and [1486] respectively.

Lorentz symmetry breaking is not the only scenario in which GWs propagate at a different speed than light. Lorentz invariant theories can exhibit such behaviour as well if they satisfy two conditions: (i) they have extra fields that are in a nontrivial configuration between the emission and the detection locations; (ii) these fields couple to metric perturbations in a way that modifies their propagation. The second conditions generally requires specific types of nonminimal coupling between the extra fields and gravity. It is satisfied by certain generalised scalar tensor theories [1424], discussed in section 2.2. Indeed, the bound on the speed has been used to obtain strong constraints on such theories as models of dark energy [670–673, 1423, 1424]. It should be stressed, however, that, due to condition (i), such constraints are inapplicable if one does not require that the scalar field be the dominant cosmological contribution that drives the cosmic speed-up. Hence, they should not be interpreted as viability constraints on the theories themselves, but only on the ability of such models to account for dark energy.

In summary, modifications in the dispersion relation of GWs can be turned into constraints on deviations from GR. How strong these constraints are depends on the theory or scenario one is considering. The clear advantage of propagation constraints is that they are straightforward to obtain and they do not require modeling of the sources in some alternative scenario. This reveals a hidden caveat: they are intrinsically conservative, as they assume that there is no deviation from the GR waveform at the source. Moreover, as should be clear from the discussion above, propagation constraints have other intrinsic limitations and can only provide significant constraints for models that lead to deviations in the dispersion relation of GWs for relatively long wavelengths. Hence, one generally expects to get significantly stronger or additional constraints by considering the deviation from the GR waveform at the source.

Contributor: K Yagi

3.2.1. Post-Newtonian parametrisation.

3.2.2. Parameterized post-Einsteinian (ppE) formalism.

In this formalism the modified waveform in the Fourier domain is given by $\tilde h (\,f) = \tilde h_{{\rm GR}}(\,f)\, {\rm e}^{{\rm i} \beta v^{2n-5}}$ [1507]. Here $\tilde h_{{\rm GR}}$ is the GR waveform, $\beta$ is the ppE parameter representing the magnitude of the non-GR deviation while $v$ is the relative velocity of the binary components. The above correction enters at nth PN order. Since n is arbitrary, this formalism can capture deviations from GR that enter at negative PN orders. It has been used in [1500] to probe strong and dynamical field gravity with GW150914 [1] and GW151226 [2]. Bounds on $\beta$ at each PN order using a Fisher analysis are shown in figure 10. Here, the authors included non-GR corrections only in the inspiral part of the waveform. Having these bounds on generic parameters at hand, one can map them to bounds on violations in fundamental pillars of GR. For example, a  −4PN deviation maps to time variation of the gravitational constant G, that allows us to check the strong equivalence principle. 0PN (2PN) corrections can be used to probe violations in Lorentz (parity) invariance.

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3.2.3. Phenomenological waveforms.

Contributor: P Pani

Depending on the physics being tested, the post-merger 'ringdown' phase of a compact-binary coalescence might be better suited than the inspiral or the merger. For instance, in the inspiral phase the nature of the binary components shows up only at high PN order (though corrections due to extra polarizations can show up at low PN order) whereas the merger phase requires time-consuming and theory-dependent numerical simulations. On the other hand, the post-merger phase—during which the remnant BH settles down to a stationary configuration—can be appropriately described by perturbation theory and it is therefore relatively simple to model beyond GR. If the remnant is a BH in GR, the ringdown phase is well-described by a superposition of exponentially damped sinusoids,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle h_+ +{\rm i} h_\times \sim \sum_{lmn} A_{lmn}(r) {\rm e}^{-t/\tau_{lmn}}\sin (\omega_{lmn} t+\phi_{lmn}), \label{ringdown} \nonumber \end{align} \tag{ 38 }$$

called quasinormal modes (QNMs) [1118, 1510–1512]. Here, $\omega_{lmn}$ is the characteristic oscillation frequency of the final BH, $\tau_{lmn}$ is the damping time, A_lmn is the amplitude at a distance r, $\phi_{lmn}$ is the phase, l and m ($|m|\leqslant l$) are angular indices describing how radiation is distributed on the final BH's sky, and n is an overtone index labeling the (infinite, countable) number of modes.

Equation (38) sets the ground for GW spectroscopy: QNMs are the fingerprints of the final BH and extracting their frequency and damping time allows for a consistency check of the GW templates [16], tests of gravity [17, 1193, 1500], and of the no-hair properties of the Kerr solution [1513]. As a corollary of the GR uniqueness theorem [1514, 1515], the entire QNM spectrum of a Kerr BH eventually depends only on its mass and spin. Thus, detecting several modes allows for multiple null-hypothesis tests of the Kerr metric [1516–1518] and, in turn, of one of the pillars of GR.

3.3.1. Background.

3.3.2. Signatures.