1 Introduction

Coincident detections of electromagnetic (EM) and gravitational-wave (GW) signals from coalescences of massive black hole binariesFootnote 1 (MBHBs) have the potential to provide unparalleled understanding of the evolution of massive black holes (MBHs) in the context of large-scale structure (Kormendy and Ho 2013; Heckman and Best 2014). Furthermore, since direct detection of GWs can yield accurate luminosity distances and EM counterparts can provide redshifts of coalescing binary systems, their combination yields fundamentally new multimessenger probes of the Universe’s expansion (Schutz 1986; Holz and Hughes 2005; Abbott et al. 2017). Coincident EM and GW detections can measure or place limits on differences in the arrival times of photons and gravitons from the same cosmological source, which would in turn inform us about the mass of the graviton and possible violations of the equivalence principle and Lorentz invariance in the gravitational sector (Kocsis et al. 2008; Hazboun et al. 2013; Yagi and Stein 2016; Abbott et al. 2016b).

The outcome of these scientific endeavors directly depends on our ability to identify observed MBHB systems characterized by both messengers. At the time of this writing such detections are anticipated but are not a foregone conclusion. Their feasibility will be determined by the properties of MBHBs and environments in which they reside, as well as by the technical capabilities of EM and GW observatories.

Coincident EM and GW detections can provide…

 • Understanding of the evolution of MBHs in the context of large-scale structure.

 • Fundamentally new probes of the universe’s expansion.

 • Unique tests of the theory of gravity.

A basic requirement for a successful detection using any messenger is that at least some fraction of galaxy mergers lead to the formation of close MBH pairs that coalesce within the age of the Universe. Observations of dual and multiple AGNs with kiloparsec separations confirm that galactic mergers are natural sites for formation of wide MBH pairs, and in some cases multiplets. Theoretical models and simulations of galactic mergers suggest that subsequent evolution of widely separated MBHs to smaller scales should be common. However, the timescales for this process remain uncertain, owing to the inherent challenges in modeling MBHB evolution from galactic to milliparsec scales. This, combined with a lack of concrete observational evidence for close, gravitationally bound MBH pairs with sub-parsec separations, currently precludes a definitive conclusion that galactic mergers result in MBHB coalescences.

The link between merging galaxies and merging MBHs is nontrivial to establish in observations, because the two events are separated by hundreds of millions to billions of years and because the MBH coalescence timescale is very short relative to all other evolutionary stages of the binary. Consequently, MBH coalescences are difficult to identify in EM observations alone, without prior knowledge about their occurrence. This is where a specialized messenger, such as GWs, will play an unparalleled role in pinpointing the instance of a MBH merger. In this case, the temporal coincidence of a prompt EM counterpart with a gravitationally timed MBH merger may offer the best chance of identifying a unique host galaxy. Combined together, the EM and GWs can provide crucial new information about the link between hierarchical structure formation and MBH growth.

Identification of a galaxy that is a host to a MBH coalescence also relies on uniqueness of the associated EM counterpart and the ability of astronomers to recognize it in observations. This entails a priori knowledge about the observational appearance of the EM counterpart as well as the wavelength band optimal for its detection. This is because there may be more than one, or indeed many, plausible host galaxies that are EM bright in the area on the sky associated with the GW signal.

Open questions

 • Do MBHBs form and coalesce and on what timescales?

 • Are there unique EM signatures associated with MBHB coalescences?

 • Do they encode the properties of MBHBs and their environments?

To be useful beyond detection, the EM and GW signatures must ultimately encode in a predictable way the information about MBHBs and environments in which they reside. If a MBHB heading for coalescence were to be detected within the next few years, the limiting factor in interpretation of its coincident GW and EM signatures would likely be the uncertainties related to its EM signatures. Indeed, the LIGO-Virgo GW community has demonstrated that mergers of stellar origin black-hole binaries can be successfully detected and interpreted even in the absence of any EM counterparts. The EM counterparts associated with MBH coalescences nevertheless remain an active and quick-paced research area with realistic potential for major breakthroughs before the first multimessenger detection is made.

The aim of this Living Review is to provide an introduction to this research topic by presenting a summary of key findings, physical processes and ideas pertaining to EM counterparts to MBH mergers as they are known at the time of this writing. We review current observational evidence for close MBHBs in Sect. 2 and discuss the relevant physical processes and timescales in Sect. 3. In Sect. 4 we summarize the possible precursor, coalescence, and afterglow EM counterparts to GWs associated with MBHBs. We describe open questions, discuss future prospects in this research area, and conclude in Sect. 5.

2 Observational evidence for close massive black-hole binaries

Galactic mergers are an important driver of galaxy evolution and a natural channel for formation of MBH pairs and, possibly, multiplets (Begelman et al. 1980). Most of the scientific attention was initially focused on the galaxies in the act of merging, and their central MBHs were considered passive participants, taken for a ride by their hosts. The realization that MBHs play an important role in the evolution of their host galaxies (Kormendy and Richstone 1995; Magorrian et al. 1998; Ferrarese and Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002) spurred a number of observational and theoretical studies, undertaken over the last few decades, with an aim to determine what happens to MBHs once their host galaxies merge.

From an observational point of view, this question is pursued through EM searches for dual and multiple MBHs with a variety of separations, ranging from \(\sim \) tens of kpc to sub-parsec scales (see De Rosa et al. 2019, for a review). The multi-wavelength searches for MBH systems with large separations, corresponding to early stages of galactic mergers, have so far successfully identified a few dozen dual and offset active galactic nuclei (AGNs; Komossa et al. 2003; Koss et al. 2011, 2016; Liu et al. 2013; Comerford et al. 2015; Barrows et al. 2016, and others). MBHs with even smaller (parsec and sub-parsec) separations are representative of the later stages of galactic mergers in which the two MBHs are sufficiently close to form a gravitationally bound pair.

Observational evidence for MBHBs (from more to less direct)

 • Emission of gravitational waves.

 • Direct imaging of double nuclei with Very Long Baseline Interferometry.

 • Do they encode the properties of MBHBs and their environments?

 • Doppler-shifted emission lines in spectra of AGNs and quasars.

A MBH pair is gravitationally bound if the amount of gas and stars enclosed within its orbit is smaller than its own mass. For a wide range of MBH masses and host galaxy properties, this happens when the two holes reach separations of \(\sim \) 1–10 pc (see Sect. 3.3). Throughout this paper, we refer to such systems as massive black-hole binaries and use the term “dual MBHs” for unbound pairs of MBHs in the same general galactic vicinity. A key characteristic of gravitationally bound MBHBs is that they are observationally elusive and expected to be intrinsically rare. While the frequency of binaries is uncertain and dependent on their unknown rate of evolution on small scales (see Sect. 3), theorists estimate that a fraction of \(<10^{-2}\) AGNs at redshift \(z<0.6\) may host MBHBs (Kelley et al. 2019b), although only a small fraction can be identified as such (see for example the estimates in Volonteri et al. 2009).

These results have two important implications: (a) any EM search for MBHBs must involve a large sample of AGNs, making the archival data from large surveys of AGNs an attractive starting point and (b) observational techniques used in the search must be able to distinguish signatures of binaries from those of AGNs powered by single MBHs. Observational techniques used to search for such systems in large archival datasets have so far largely relied on direct imaging, photometry, and spectroscopic measurements. They have recently been complemented by observations with pulsar timing arrays (PTAs), which could detect GWs from massive MBHBs. In the remainder of this section we summarize the outcomes of and future prospects for these different observational approaches. We also direct the reader to Dotti et al. (2012) and Schnittman (2013) for reviews of a broad range of MBHB signatures proposed in the literature.

2.1 Emission of gravitational waves

Detection of gravitational radiation emitted by a MBHB heading for coalescence constitutes the most direct evidence for the existence of such systems. At the time of this writing, GWs from MBHBs have not yet been definitively detected. However, expectations for such detections of gravitational radiation have been raised by the success of the LIGO-Virgo Collaboration (e.g., Abbott et al. 2016a), by the selection of the Laser Interferometer Space Antenna (LISA) for a large-class mission in the European Space Agency science program, and by the success of the LISA Pathfinder mission (Armano et al. 2016) and the TianQin-1 experimental satellite (Luo et al. 2020). Recently, three PTA experiments, the North American Nanohertz Observatory for Gravitational Waves (NANOGrav), the Parkes Pulsar Timing Array (PPTA) and the European Pulsar Timing Array (EPTA) reported strong evidence for a signature of the common-spectrum, stochastic process (Arzoumanian et al. 2020; Goncharov et al. 2021; Chen et al. 2021). Even though the signal does not show sufficient evidence of quadrupolar correlations needed to claim detection of a stochastic background of GWs from MBHBs (see Fig. 1 and the text below), it provides a useful testing ground for a variety of theoretical models, and adds to the sense that an era of low-frequency GW astronomy is imminent.

Fig. 1
figure 1

Image(s) reproduced with permission from [top] Aggarwal et al. (2019), and [bottom] Arzoumanian et al. (2020), copyright by AAS

Top: Sky-averaged sensitivity (shown as the upper limit on the GW strain amplitude) as a function of the GW frequency from the NANOGrav 5-year data set (green), 9-year data set (orange), and 11-year data set (blue). The data is presently most sensitive at the frequency of 8 nHz and least sensitive around \(3\times 10^{-8}\,{\mathrm{Hz}} = 1\,{\mathrm{yr}}^{-1}\), corresponding to the orbital frequency of Earth around the Sun. Any MBHBs with GW strain above the 11-year sensitivity curve would at this point be detected by NANOGrav. Bottom: Angular correlation of common-spectrum pulsar timing noise versus angular separation of pulsars on the sky, from the NANOGrav 12.5-year dataset. The violin plots (gray/black) show the marginalized posteriors for the interpulsar spatial correlations. For comparison, the orange dashed line shows the flat correlation signature between pulsars expected as a signature of error in the timescale (e.g., drifts in clock standards), and the blue dashed curve shows the Hellings & Downs correlation predicted to result from a stochastic background of GWs from MBHBs.

PTAs seek to detect GWs by searching for correlations in the timing observations of a network of millisecond pulsars (Hellings and Downs 1983; Foster and Backer 1990). Specifically, the deviation of pulse arrival times from different pulsars due to an intervening nHz GW is predicted to exhibit a characteristic correlation with the angle between the pulsars on the sky; this is the so-called Hellings and Downs (1983) curve. The three mentioned PTA experiments currently in operation, NANOGrav (McLaughlin 2013), EPTA (Desvignes et al. 2016), and PPTA (Hobbs 2013), together form the International Pulsar Timing Array (IPTA; Verbiest et al. 2016). PTAs are sensitive to GWs with frequencies between a few and few hundred nHz, and their sensitivity increases for longer observational time baselines. This is illustrated in Fig. 1, which shows the sensitivity as function of the GW frequency of NANOGrav based on the 5, 9 and 11 years of timing observations of about forty pulsars (Aggarwal et al. 2019). PTA sensitivity is also being steadily improved as more pulsars are identified and added to the network.

The expected signals for PTAs include (i) the aforementioned stochastic GW background, which is produced by an ensemble of unresolved MBHBs emitting GWs at different frequencies and distances throughout the Universe, and (ii) continuous GWs from individual MBHBs, which would exhibit negligible frequency evolution on human timescales. At this point it is unclear whether the stochastic GW background will be detected first (Rosado et al. 2015; Arzoumanian et al. 2020) or whether it will be detected contemporaneously with the individual MBHBs (Mingarelli et al. 2017; Kelley et al. 2018). In either case, a wide range of theoretical predictions agree that GWs from MBHBs are expected to be observed by the PTAs within the next decade. If so, their discovery and analysis will be well timed to inform expectations for GW detections with the space-based observatory LISA, which is planned to launch in the mid-2030s. Because PTAs are sensitive to the most massive MBHB systems out to \(z\sim 2\) (Sesana et al. 2008; Chen et al. 2017), they stand a chance of being associated with some of the most EM luminous AGNs in the local Universe. This bodes well for the prospects for multimessenger studies of MBHBs in the near future.

PTAs are sensitive to individual MBHBs with high total mass, \(\sim 10^8\)\(10^{10}\,M_\odot \), and orbital periods on the order of months to years. For example, the NANOGrav collaboration performed a search in their 11-year dataset for GWs from individual MBHBs on circular orbits. While no statistically significant evidence for GWs has been found in the data, a physically interesting limit has been placed on the distances to, or equivalently masses of, individual MBHBs (Aggarwal et al. 2019). The 11-year data can rule out the existence of MBHBs within 120 Mpc with chirp massFootnote 2\(\mathscr {M} = 10^9\,M_\odot \) emitting GWs with frequency 8 nHz, which falls in the most sensitive part of the spectrum (see Fig. 1). The existence of binaries with even larger chirp masses of \(\mathscr {M} = 10^{10}\,M_\odot \), emitting at the same frequency, can be ruled out to even higher, cosmological distances of 5.5 Gpc.

In contrast to PTAs, the planned LISA and TianQin space-based missions will detect higher GW frequencies and MBHBs with lower total masses (Amaro-Seoane et al. 2017; Mei et al. 2021). LISA will be sensitive to GWs with frequencies in the range from about \(100\,\mu \)Hz to 100 mHz. The frequency of GW radiation emitted by MBHBs with total masses of \(10^4\)\(10^7\,M_\odot \) falls squarely within this bandwidth in the last stages of their evolution (Klein et al. 2016). Figure 2 illustrates that for many binary configurations in this mass range, high signal-to-noise merger detections will be possible out to redshift of \(z\sim 20\), which will in turn allow measurements of the masses and spins of coalescing MBHs to a few \(\%\) accuracy. LISA will therefore be able to place the evolution of MBHs in the context of important cosmic epochs that coincide with the formation of MBH seeds (marked as Cosmic Dawn in Fig. 2), the most distant quasars (Reionization), and the epoch of growth through accretion and mergers (Cosmic Noon). TianQin will operate over a similar (albeit not identical) parameter space and, like LISA, is expected to detect GWs from actual MBHB merger events (Feng et al. 2019; Wang et al. 2019). Both in terms of the mass range and redshift distribution, the space-based GW observatories will therefore unearth a population of MBHBs that is different from and complementary to those discovered by PTAs.

Fig. 2
figure 2

Image(s) reproduced with permission from Colpi et al. (2019)

Contours of constant signal-to-noise ratio for the LISA observatory plotted as a function of redshift (or equivalently, cosmic time) and MBHB mass. The contours are calculated for systems of non-spinning MBHs with mass ratio \(q=0.5\). Overlaid is the illustration of an evolutionary track, which ends with a formation of a \(\sim 10^7\,M_\odot \) MBH merger remnant at \(z\approx 3\). Black dots and arrows represent the MBHs and their spins, respectively. MBHs are embedded in galactic halos (illustrated by white-yellow circles) and grow through episodes of accretion (black lines) and mergers. Black stars mark the redshifts of the most distant quasar, long gamma-ray burst and galaxy detected so far.

From all MBHBs detected with the space-based GW observatories, the population that stands the best chance of also being detected in EM observations includes systems at intermediate and low redshifts. Figure 3 shows the tracks for several representative MBHBs with equal mass ratios at redshift \(z=3\) as they evolve through the LISA frequency band. MBHBs with mass \(10^5\)\(10^7\,M_\odot \) merging in galaxies at \(z \le 4\) are sufficiently loud GW sources to be detected by LISA weeks before coalescence. For systems in this mass and redshift range, the sky localization determined from the GW observations alone is expected to reach accuracy of about 10–\(100\,{\mathrm{deg}}^2\) in the weeks to hours before coalescence and \(\approx 0.1\,{\mathrm{deg}}^2\) at merger (Haiman 2017; Mangiagli et al. 2020). As long as these systems are also sufficiently EM luminous and can maintain emission regions until very close to the coalescence, this localization accuracy provides an opportunity for subsequent identification of an EM counterpart to GW detection. For example, if the culprit source (an AGN on the sky) exhibits characteristic EM variability that correlates with the GW chirp, this could lead to a convincing EM identification of the binary headed for coalescence. Under such conditions, Dal Canton et al. (2019) find that given a fiducial LISA detection rate of 10 mergers per year at \(z<3.5\), a few detections of modulated X-ray counterparts are possible over the nominal, 5 year duration of the LISA mission, with an X-ray telescope with a relatively large (1 \(\hbox {deg}^2\)) field of view. Whether MBHBs have unique EM signatures in this or any other stages of their evolution that can be distinguished from regular AGNs is presently a subject of active investigation.

Fig. 3
figure 3

Image(s) reproduced with permission from the LISA L3 mission proposal (Amaro-Seoane et al. 2017)

Illustration of the evolutionary tracks (multicolor) through the LISA frequency band for three equal mass MBHBs at redshift \(z=3\) with total intrinsic masses \(10^5\), \(10^6\) and \(10^7\,M_\odot \). The remaining time before coalescence is marked on the tracks. Also shown are the signals from the extreme mass-ratio inspiral (EMRI), stellar origin black-hole binaries of the type discovered by the LIGO-Virgo Collaboration, and from resolved galactic binaries (including the already known verification binaries). The solid green curve shows LISA’s sensitivity and black dashed line is the sensitivity with an additional confusion signal at \(\sim 1\) mHz contributed by the unresolved galactic binaries. The sensitivity curves correspond to a configuration with six links, 2.5 million km arm length and a mission lifetime of 4 years.

2.2 Direct imaging of double nuclei

A clear EM manifestation of a parsec-scale MBHB is an image of a binary AGN that forms a gravitationally bound system (as opposed to an accidental projection on the sky). A practical obstacle in the detection of such objects arises from their small angular separation: for example, a parsec-scale binary at a redshift of \(z=0.1\) subtends an angle of only \(\sim 0.5\) mas on the sky (neglecting the projection effects). Such scales are below the angular resolution of most astronomical observatories operating at present time, except the radio and millimeter observatories using the Very Long Baseline Interferometry (VLBI) technique (e.g., D’Orazio and Loeb 2018).

This approach has been used to identify the most convincing MBHB candidate thus far, in the radio galaxy 0402+379 (Rodriguez et al. 2006, 2009; Morganti et al. 2009). This system hosts a pair of two compact radio cores at a projected separation of 7.3 pc on the sky (Fig. 4). Both cores are characterized by flat radio spectra and have been identified as possible AGNs based on this signature (Maness et al. 2004; Rodriguez et al. 2006). Bansal et al. (2017) subsequently reported that long term VLBI observations reveal relative motion of the two cores, consistent with orbital motion, lending further support to the MBHB hypothesis. While the MBHB candidate in the galaxy 0402+379 was discovered serendipitously, it demonstrated the power of radio interferometry in imaging of small-separation MBHs.

Fig. 4
figure 4

Image(s) reproduced with permission Rodriguez et al. (2006), copyright by AAS

The MBHB candidate in the radio galaxy 0402+379. Two compact radio cores in the middle of the figure are separated by 7.3 pc on the sky and are thought to harbor MBHs. Both cores are characterized by flat radio spectra and have been identified as possible AGNs based on this signature. Radio lobes from a jet associated with one of the cores are also apparent. The color illustrates the spectral index distribution between 8 and 22 GHz from the VLBA observations. The inset panels show flux density as a function of the logarithm of frequency for each core.

In a subsequent investigation, Burke-Spolaor (2011) searched for binaries in the archival VLBI data. The search targeted spatially resolved, double radio-emitting nuclei with a wide range of orbital separations (\(\sim 3\,\mathrm{pc}\)\(5\,\mathrm{kpc}\)) among 3114 radio-luminous AGNs in the redshift range \(0 < z \le 4.715\). Another investigation, in addition to MBHBs, searched for the recoiling MBHs spatially offset from the centers of their host galaxies (Condon et al. 2011). The latter study is based on Very Long Baseline Array (VLBA) observations at a frequency of 8 GHz of a sample of 834 nearby radio-luminous AGNs with typical distances of \(\sim 200\) Mpc. Radio sources brighter than 100 mJy were selected from the 1.4 GHz NVSS catalogue data, under the hypothesis that the most massive galaxies have undergone a major merger in their history and may therefore contain a binary or recoiling MBH. More recently, Tremblay et al. (2016) performed multi-frequency imaging of a flux-limited sample of \(\sim \)1100 AGN included in the VLBA Imaging and Polarimetry Survey, all of which were pre-selected to be bright and flat-spectrum. None of these searches unearthed new instances of double radio nuclei, leaving several possible interpretations: (a) there is a true paucity of MBHBs, (b) MBHBs may be present but have low radio brightness, or (c) only one component of the binary is radio-bright and it may or may not show a detectable spatial offset relative to the center of the host galaxy. This highlights the difficulties in using radio imaging as the primary technique for selecting gravitationally bound MBHB candidates or their progenitors, given their unknown radio properties.

Notwithstanding the challenges, VLBI imaging still represents a powerful technique. One promising avenue for VLBI may be to test candidate MBHBs selected by other techniques, especially if longer baseline arrays operating at higher frequencies become available in the future. Figure 5 summarizes the capabilities of one such array, proposed as an extension of the Very Large Array (VLA), and named the Next Generation VLA (ngVLA; Murphy 2018; Burke-Spolaor et al. 2018). The ngVLA could resolve gravitationally bound binaries at \(z\lesssim 0.1\) when operating at 10 GHz, with the main array baseline of 1000 km and subarrays extending to \(\sim 9000\) km used in VLBI mode. The same array configuration operating at 120 GHz could in principle resolve MBHBs at even higher redshifts, as well as those that are inspiraling due to the emission of GWs at \(z\lesssim 0.01\). Therefore, VLBI searches with longer baselines and higher frequencies have the potential to enable both multi-wavelength and multimessenger searches for MBHBs at a variety of separations. If funded, the construction of the ngVLA could commence as early as 2024 with full operations starting in 2034, enabling it to operate contemporaneously with another next generation radio observatory currently under development in the Southern hemisphere, the Square Kilometer Array,Footnote 3 as well as with the low-frequency GW detectors described in the previous section.

Fig. 5
figure 5

Image reproduced with permisson from Burke-Spolaor et al. (2018), copyright by ASP

Projected nuclear separation as a function of redshift for the observed population of candidate AGN pairs and binaries, compared with the spatial resolution of the proposed ngVLA observatory. Orbital periods associated with each nuclear separation have been calculated neglecting the orbital projection effects. Each symbol represents a candidate discovered in observations in the radio (circles), optical / near-IR (dots), or X-ray (\(\times \)) part of the spectrum. Horizontal lines mark approximate evolutionary stages of MBH pairs and binaries. The black curve marks the spatial resolution limit at 10 GHz of the VLA observatory in the A-configuration. The red and purple curves indicate the spatial resolution limit in the 10 GHz and 120 GHz bands, respectively, for the ngVLA 150 km baseline array (dashed) and 1000 km extended-baseline array with VLBI capabilities (solid). Longer baselines and higher frequencies have the potential to resolve MBHBs in the gravitational-wave regime in the nearby universe (indicated by the curves that cross the dark yellow region).

2.3 Photometric measurements of quasi-periodic variability

MBHBs at \(\sim \) mpc-scale separations have orbital periods of \(\sim \) years, making any associated EM variability accessible on human timescales. Many MBHB candidates have been identified via signatures of periodic or nearly periodic variability in the lightcurves of quasars, which are interpreted as a manifestation of binary orbital motion. The possible physical origins of periodic EM variability from MBHBs are discussed in Sect. 4.1.1. A well-known example of a MBHB candidate in this category is the blazar OJ 287, which exhibits outburst activity in its optical light curve with an observed period close to 12 years (see Fig. 6 from Valtonen et al. 2008). It is worth noting however that OJ 287 is unique among photometrically selected binary candidates because the first recorded data points in its light curve extend as far back as the 19th century. Similarly, OJ 287 has received an unprecedented level of observational coverage in modern times, yielding a high frequency of sampling (from 1970s onwards), in addition to long observational baseline. While the presence of quasi-periodicity has been claimed in other objects, it is usually less pronounced and recorded over much shorter time span than in the case of OJ 287, thus preventing an ironclad case for existence of MBHBs from being made (for e.g., Fan et al. 1998; Rieger and Mannheim 2000; De Paolis et al. 2002; Liu et al. 2014a). In 2019, a bright flare was observed from OJ 287 that occurred within 4 hours of its predicted arrival time, which provides further support for the MBHB hypothesis for this source (Laine et al. 2020).

Fig. 6
figure 6

Image reproduced with permisson from Valtonen et al. (2008), copyright by NPG

Historical light curve of OJ 287 in V-magnitude recorded over more than 100 years. Quasi-periodic outbursts with an observed period of \(\sim 12\) yr in the light curve of this object have been interpreted as a signature of the MBHB orbital motion.

Recently, there have been a number of systematic searches for MBHBs in photometric surveys, such as the Catalina Real-Time Transient Survey, the Palomar Transient Factory and others (e.g., Graham et al. 2015a; Charisi et al. 2016; Liu et al. 2016). Although some candidates have been identified, there is an ongoing discussion about the level of evidence provided by the data. For example, as pointed out by Vaughan et al. (2016) and later demonstrated by Liu et al. (2018), the risk of false binary identifications in photometric surveys is especially high when the evidence is based on only a few apparent orbital cycles. If the current candidate fraction is treated as an upper limit, then the incidence of MBHBs among quasars is \(< 5 \times 10^{-4}\), which is consistent with theoretical predictions for low redshift (\(<10^{-2}\); Volonteri et al. 2009; Kelley et al. 2019b).

Because of the finite temporal extent of the surveys, which must record at least several orbital cycles of a candidate binary, most photometrically identified MBHB candidates have relatively short orbital periods, of the order a few years or less. While systematic searches like these signal more efficient approach to finding MBHB candidates in surveys, a confirmation of their identity requires additional evidence. This is because stochastically variable lightcurves of “normal” AGNs and quasars, powered by single MBHs, can be mistaken for periodic sources.

Photometric searches for MBHBs will expand greatly in the next decade, as the capabilities of the optical time-domain surveys will be dramatically enhanced with the Zwicky Transient Facility (ZTF) and the advent of the Legacy Survey of Space and Time (LSST) with the Vera Rubin Observatory in 2023. Over time, photometric searches for MBHBs will also become increasingly sensitive to a wider range of orbital periods, as the existing surveys continue to collect data on longer timescales. For example, Kelley et al. (2019b) find that if the host galaxy does not outshine the AGN, up to \(\sim 100\) such MBHB candidates could be identifiable after 5 years of LSST observations with the Rubin Observatory. They also suggest existence of a “sweet spot” for binary mass ratios of \(q \sim 0.1\), where differential accretion onto the secondary MBH could enhance variability. This could potentially improve the prospects for observing some MBHB systems with lower total mass. Xin and Haiman (2021) furthermore predict that from all quasi-periodic AGNs detected by LSST (of which there may be 20 to 100 million), there will be 10–150 powered by MBHBs with short orbital periods (\(\lesssim 1\)  day) that could in ~ 5–15 years be detected by LISA. Such sources would thus serve as “LISA verification binaries", analogous to short-period Galactic compact-object binaries.

Already, some multimessenger constraints have been placed on the MBHB population by combining these photometrically variable candidates with upper limits from PTAs on their GW signals. PTAs are capable of probing the GW background at nanoHertz frequencies from sources such as MBHBs with orbital periods of a few years (Lentati et al. 2015; Shannon et al. 2015; Arzoumanian et al. 2016, 2020). Specifically, the upper limit placed by the sensitivity of PTAs largely rules out the amplitude of GW background resulting from the so far identified \(\sim 150\) photometric binary candidates, implying that some fraction of them are unlikely to be MBHBs (Sesana et al. 2018). Subsequently, a similar approach was used to place limits on the presence of MBHBs in periodic blazars (Holgado et al. 2018) and in ultraluminous infrared galaxies (Inayoshi et al. 2018). More recently, Liu and Vigeland (2021) illustrated how detection and parameter estimation of individual MBHBs in PTA data could be improved by up to an order of magnitude with the addition of EM priors. These studies provide examples of the great potential of multimessenger techniques, even prior to a low-frequency GW detection.

2.4 Spectroscopic measurements of offset broad emission lines

The principal assumption made by all spectroscopic searches is that some fraction of MBHBs at sub-parsec orbital separations reside in emission regions comparable in size to the broad line regions (BLRs) of regular AGNs.Footnote 4 If so, the dynamical perturbation of a BLR by the binary gravitational potential can in principle be reflected in the low-ionization broad emission-line profiles (e.g., Gaskell 1983, 1996; Bogdanović et al. 2008; Montuori et al. 2011, 2012). The broad emission lines of particular interest are H\(\alpha ~\lambda 6563\), H\(\beta \;\lambda 4861\) and Mg ii\(\;\lambda 2798\), because they are prominent in AGN spectra and are commonly used as tracers of dense, low-ionization gas in BLRs at low (H\(\alpha \) at \(z < 0.4\)) and high redshift (Mg ii at \(z < 2.5\)).

Spectroscopic searches rely on the detection of the Doppler shift of broad emission lines in the spectrum of a MBHB host caused by the binary orbital motion. This approach is reminiscent of a well established technique for detection of single- and double-line spectroscopic binary stars. In both classes of spectroscopic binaries, the lines are expected to oscillate about their local rest frame wavelength on the orbital time scale of a system. In the context of the MBHB model, the spectral emission lines are assumed to be associated with gas accretion disks that are gravitationally bound to the individual black holes (Gaskell 1983, 1996; Bogdanović et al. 2009a). Given the high velocity of the bound gas, the emission-line profiles from the MBH disks are expected to be Doppler-broadened, similar to the emission lines originating in the BLRs of regular AGNs. Moreover, several theoretical studies have shown that in unequal-mass binaries, accretion occurs preferentially onto the lower mass object (Artymowicz and Lubow 1994; Günther and Kley 2002; Hayasaki et al. 2007; Roedig et al. 2011; Farris et al. 2014), rendering the secondary AGN potentially more luminous than the primary. If so, this indicates that some fraction of MBHBs may appear as single-line spectroscopic binaries. This scenario is illustrated in Fig. 7, where the Doppler-shifted broad component of the H\(\beta \) emission line traces the motion of the smaller MBH. (See however Nguyen et al. 2020a, who suggest that the motion traced may actually be that of the primary).

Fig. 7
figure 7

Image(s) reproduced with permission from Guo et al. (2019), copyright by the authors

Illustration of the spectroscopic radial velocity technique used to search for MBHBs. In the context of the MBHB model adopted by spectroscopic surveys, the broad profile (shown in the top panels as the H\(\beta \) line) is attributed to the emission from the accretion disk of the smaller MBH. The offset of the broad emission-line profile with respect to its rest frame wavelength (marked by the vertical dotted line) is attributed to the binary orbital motion. The center of mass of the MBHB is assumed to be at rest with respect to the host galaxy, and the rest frame of the host galaxy is defined by the narrow H\(\beta \) and [O \(\textsc {iii}\)] lines.

This realization led to the discovery of a number of MBHB candidates based on the criterion that the culprit sources exhibit broad optical lines offset with respect to the rest frame of the host galaxy (Bogdanović et al. 2009b; Dotti et al. 2009a; Boroson and Lauer 2009; Tang and Grindlay 2009; Decarli et al. 2010a; Barrows et al. 2011; Tsalmantza et al. 2011; Eracleous et al. 2012; Tsai et al. 2013).Footnote 5 Because this effect is also expected to arise in the case of a recoiling MBH receding from its host galaxy, the same approach has been used to flag candidates of that type (Komossa et al. 2008; Shields et al. 2009b; Civano et al. 2010; Robinson et al. 2010; Lusso et al. 2014). The key advantage of the method is its simplicity, as the spectra that exhibit emission lines shifted relative to the galaxy rest frame are fairly straightforward to select from large archival data sets, such as the Sloan Digital Sky Survey (SDSS). Its main complication, however, is that the Doppler-shift signature is not unique to these two physical scenarios; for example, broad line offsets can also be produced by AGN outflows. Complementary observations are therefore required in order to determine the nature of the observed candidates (e.g., Popović 2012; Barth et al. 2015).

To address this ambiguity, a new generation of time-domain spectroscopic searches has been designed to monitor the offset of the broad emission-line profiles over multiple epochs and identify sources in which modulations in the offset are consistent with binary orbital motion (Bon et al. 2012, 2016; Eracleous et al. 2012; Decarli et al. 2013; Ju et al. 2013; Liu et al. 2014b; Shen et al. 2013; Runnoe et al. 2015, 2017; Li et al. 2016; Wang et al. 2017; Guo et al. 2019). For example, Eracleous et al. (2012) searched for \(z < 0.7\) SDSS AGNs with broad H\(\beta \) lines offset from the rest frame of the host galaxy by \(\gtrsim 1000\,\mathrm{km\,s^{-1}}\). Based on this criterion, they selected 88 quasars for observational follow-up from an initial group of about 16,000. The follow-up observations of this campaign span a temporal baseline from a few weeks to 12 years in the observer’s frame. After multiple epochs of follow-up, statistically significant changes in the velocity offset have been measured for 29/88 candidates (Runnoe et al. 2015, 2017), all of which remain viable as MBHB candidates. This implies that \(\lesssim 10^{-3}- 10^{-2}\) of quasars host MBHBs at \(z < 0.7\), a limit that is so far consistent with theoretical predictions (Volonteri et al. 2009; Kelley et al. 2019b).

Figure 8 shows several representative broad \(H\beta \) profiles selected in this search. The profiles are asymmetric and have peaks offset by \(\sim {\mathrm{few}}\times 10^3\;\mathrm{km\,s^{-1}}\) from the rest frame of the galaxy, inferred from the wavelength of the narrow emission lines (the narrow emission-line components were subsequently removed from these profiles for clarity). If measured velocity offsets of this magnitude are associated with the orbital motion of the secondary MBH, as described above, then it is possible to show that searches of this type (with a yearly cadence of observations) are in principle sensitive to a subset of MBHBs with orbital separations \(\lesssim {\mathrm{few}}\times 10^4\) Schwarzschild radii and orbital periods in the range of a few tens to a few hundreds of years (Pflueger et al. 2018). In comparison, current spectroscopic observational campaigns typically span a baseline about 10–15 years, shorter than the average period of MBHBs targeted by this technique.

Fig. 8
figure 8

H\(\beta \) emission-line profiles of MBHB candidates selected in a spectroscopic survey by Eracleous et al. (2012). Only the broad emission-line profiles are shown; the narrow line components were removed for clarity. Peaks of the asymmetric profiles are offset from the rest frame of the host galaxy (marked by the dashed vertical line) by \(\sim {\mathrm{few}}\times 10^3\;\mathrm{km\,s^{-1}}\). If measured velocity offsets are associated with the orbital motion of the secondary MBH, they imply orbital periods in the range of \(\sim {\mathrm{few}}\times 10- {\mathrm{few}}\times 100\) yr for MBH mass ratios larger than 0.1. The dotted lines mark velocity offsets of \(\pm 5000\;\mathrm{km\,s^{-1}}\)

These relatively short monitoring baselines still allow one to reject MBHB candidates with velocity curves inconsistent with the MBHB model, even if they have been monitored for less than a full orbital cycle (Runnoe et al. 2017; Guo et al. 2019). At the same time, they preclude measurement of multiple orbital cycles, which have traditionally been used as a criterion for binarity in stellar systems. Therefore, this technique alone cannot uniquely confirm the identity of a MBHB; it must be supplemented by additional, independent observational evidence that can help to further elucidate their nature. Some of the more promising complementary approaches include direct VLBI imaging at millimeter and radio wavelengths of nearby MBHB candidates (\(z\lesssim 0.1\)) with orbital separations \(a \gtrsim 0.01\,\)pc (e.g., D’Orazio and Loeb 2018; Burke-Spolaor et al. 2018; Breiding et al. 2021).

As in the case of the photometric searches, a statistical test of the nature of spectroscopic MBHB candidates can be performed using GW constraints from PTAs. While hypothetical MBHBs detected spectroscopically do not emit significant amounts of GWs because they are at relatively large separations, they imply some number of MBHBs inspiraling toward coalescence, whose GW signal is reaching Earth at this very moment. For example, a comparison of this GW signal with the PTA sensitivity limit indicates that all the MBHB candidates in the spectroscopic sample of Eracleous et al. (2012) could be binaries without violating the observational constraints on the GW background (Nguyen et al. 2020b).

The risk of binary misidentifications in spectroscopic surveys can also be mitigated by continued monitoring and longer observational baselines. One example of a survey that can provide such opportunity in the next decade is SDSS-V, which is planned for the period 2020–2025.Footnote 6 As a part of SDSS-V, the Black Hole Mapper multi-object spectroscopic survey will provide multi-epoch optical spectra for more than 400,000 X-ray sources to probe the evolution of MBHs over cosmic time. This and similar surveys will therefore provide a considerable sample of active galaxies that are potential hosts to MBHBs.

3 Physical processes and relevant timescales

3.1 Rates and typical redshifts

In the standard model of structure formation in the Universe, bound halos of dark matter (along with their associated baryons) assemble in a hierarchical fashion, with small structures forming first and combining to make larger structures. If halos contain central MBHs, then halo mergers can lead to eventual mergers of the black holes. Such a coalescence will typically not happen if one halo is much less massive than the other, because in that case dynamical friction is inefficient and the lower-mass halo will either have an inspiral time longer than the Hubble time, or it will dissolve before it settles in the center. Indeed, Taffoni et al. (2003) find that below mass ratios of 1:10, merging of dark matter halos is ineffective. As we discuss in the next section, a similar result is found in simulations of isolated (i.e., non-cosmological) galaxy mergers and cosmological simulations that follow pairing of MBHs.

Analyses of halo mergers (Press and Schechter 1974; Lacey and Cole 1993; Sheth et al. 2001; Menou et al. 2001; Volonteri et al. 2002, 2003; Wyithe and Loeb 2003; Sesana et al. 2004; Yoo and Miralda-Escudé 2004; Islam et al. 2004a, b; Shapiro 2005; Sesana et al. 2005; Micic et al. 2007, and others) indicate that most of the halo mergers that might lead to MBH mergers occur at redshifts of several. One therefore finds that, if they follow promptly after their host halos merge, most MBH mergers would happen at redshifts \(z>3\) and perhaps at even higher redshifts. Such mergers would be difficult to detect electromagnetically because of their distance and the weakening of the signal due to redshift, and it will be even more difficult to identify their host galaxies. There is however evidence that delays due to orbital evolution of MBHs may shift the peak of the MBH coalescence rate to \(z\sim 1-2\), a much more promising outcome for the EM followup (Volonteri et al. 2020; Li et al. 2022). The space-based gravitational-wave detectors (like LISA and TianQin) will nonetheless be able to detect mergers out to \(z\approx 20\) and will therefore provide unique insight into the era of structure assembly.

Physical mechanisms leading to orbital evolution of MBHBs

 • Dynamical friction (during pairing of MBHBs).

 • Stellar “loss-cone” scattering.

 • Interactions with additional MBHs.

 • Torques from a circumbinary gas disk.

 • Emission of gravitational waves.

As discussed in Sect. 2.1, EM counterparts to MBH mergers would be much easier to detect at redshifts \(z<1\). The rate of such mergers, which may be accompanied by EM counterparts, is difficult to determine directly, but if we assume that major galaxy mergers (with mass ratios \(>1/4\)) inevitably lead to MBH mergers, then because galaxies comparable to or more massive than the Milky Way have typically had at least one major merger since \(z\sim 1\) (Bell et al. 2006; Lotz et al. 2008; López-Sanjuan et al. 2009; Bundy et al. 2009; Hopkins et al. 2010), and there are tens of billions of such galaxies in the universe, one might expect a few such mergers to happen per year.

In the last few decades it has become clear that the mass of a central MBH is correlated with properties of its host galaxy (e.g., Kormendy and Richstone 1995; Magorrian et al. 1998; Ferrarese and Merritt 2000; Gebhardt et al. 2000; Merritt and Ferrarese 2001; Tremaine et al. 2002; Ferrarese 2002; Ferrarese et al. 2006; Gültekin et al. 2009; McConnell and Ma 2013). Perhaps the tightest correlation is between the black hole mass, M, and the stellar velocity dispersion of the bulge, \(\sigma \). A recent assessment of this correlation (McConnell and Ma 2013) gives

$$\begin{aligned} \log (M/M_\odot )=\alpha +\beta \log (\sigma /200~\mathrm{km~s}^{-1}) \,, \end{aligned}$$

with \(\alpha =8.32\pm 0.05\), \(\beta =5.64\pm 0.32\), and there is an intrinsic scatter in the logarithm of the MBH mass of \(\epsilon _0=0.38\). This correlation implies that moderate to massive galaxies typically contain black holes with masses \(M\sim 10^6\)\(10^8\,M_\odot \). Mergers between such black holes release energy in GWs that is comparable to the total energy emitted by the stars in the galaxy over a Hubble time. As we discuss later in this section, some of that energy is emitted asymmetrically and hence, the remnant can receive a kick sufficient to eject it from the merged host galaxy.

Characteristic timescales

 • Galaxies merge in a few Gyr.

 • Bound MBHB forms in \(\sim 10^{7-9}\,{\mathrm{yr}}\) due to dynamical friction.

 • MBHB shrinks in \(\sim 10^{6-9}\,{\mathrm{yr}}\)

 • On the smallest scales MBHB coalesces rapidly due to the emission of GWs.

There are several important stages between a galaxy merger and a black hole merger, defined in terms of the physical mechanisms that determine the orbital evolution of the MBH pair. In brief:

  1. 1.

    The interacting galaxies settle together in a few orbital times. The bulges of the galaxies, which contain the MBHs, and are several hundred times more massive than the holes, are initially dense enough to act dynamically as single objects. They therefore sink in the overall star-gas system of the remnant galaxy. This can take a few Gigayears.

  2. 2.

    When the cores get within several hundred parsecs of each other, they interact dynamically and the MBHs shed their stars. The holes now spiral separately in the potential of the remnant galaxy, and their orbit evolves due to dynamical friction against the stars and gas. This stage can take between \(\sim 10^7\) to \(10^9\,{\mathrm{yr}}\).

  3. 3.

    When the MBHs get close enough so that their total mass significantly exceeds the mass of the gas and stars within their orbits (typically, \(\sim \) few pc), their orbital evolution can slow, or even stall in principle, if much of the interacting stars and gas are ejected. As a consequence, the MBHB evolution in this stage is a sensitive function of the properties of its environment and spans a wide range of timescales (\(\sim 10^6\)\(10^9\,{\mathrm{yr}}\)).

  4. 4.

    If the holes can bridge this gap and reach separations of \(10^{-2}\)\(10^{-3}\,\mathrm{pc}\), then gravitational radiation by itself suffices to bring the MBHs to coalescence in less than a Hubble time. This phase can be very short, compared to the the other evolutionary stages mentioned above.

The last three evolutionary stages are illustrated in Fig. 9. We now discuss them in more detail, in the context of stellar and gas dynamical processes.

Fig. 9
figure 9

Image(s) reproduced with permission from Begelman et al. (1980), copyright by Macmillan

Diagram of timescales associated with different stages of MBHB evolution: dynamical friction (\(t_{\mathrm{df}}\)), loss-cone scattering (\(t_{\mathrm{h}}\)), gas processes (\(t_{\mathrm{gas}}\)), and GW emission (\(t_{\mathrm{GR}}\)). The timescales correspond to a relatively massive binary, \(M \approx 10^8\,M_\odot \), residing in a massive elliptical galaxy with stellar velocity dispersion \(\sigma = 300\,\mathrm{km\,s^{-1}}\). Unless the loss cone can be replenished on a timescale much less than the stellar relaxation time (dotted line), the MBHB orbital evolution takes longer than a Hubble time (solid). Gas dynamical processes can in principle lead to faster MBHB evolution (short dashed). For binaries that reach smaller separations, gravitational radiation takes over as the dominant mechanism at \(\sim 10^{-2}\,\)pc (long dashed).

3.2 Effects before MBHs are bound to each other

3.2.1 Stellar and gas dynamical friction

Dynamical friction is thought to be the principal mechanism responsible for orbital evolution of MBHs in the aftermath of galactic mergers and an important channel for formation of gravitationally bound MBH binaries. When two galaxies merge, their MBHs are initially tightly bound to their individual stellar bulges, so that the bulge-MBH system dynamically acts as one object. The parent bulges continue to sink towards each other until they reach \(\sim 1\,\)kpc. Beyond this point the bulges merge and gradually dissipate until the MBHs are on their own. As long as the mass of the stars and gas between the MBHs is significantly larger than the total mass of the holes, the MBHs are dragged towards each other by dynamical friction exerted by the background stars and gas in the newly-formed remnant galaxy. Thus, for a wide range of galactic merger scenarios, this process is expected to play an important role for MBHs with separations in the range from \(\sim {\mathrm{few}}\times 100\,\)pc to \(\sim {\mathrm{few}}\,\)pc, when they become gravitationally bound. Dynamical friction is also expected to be important for MBHs that are ejected to comparable distances in dynamical interactions with other MBHBs, or in the aftermath of the GW recoil, the effects that we discuss in Sects. 3.3.1 and 3.4.

In this section we summarize the key aspects of dynamical friction in stellar and gaseous media, as applicable to MBHs in those scenarios.Footnote 7 For a general introduction to this topic, we direct the reader to the seminal work of Chandrasekhar (1943), who developed the analytic theory for gravitational drag in collisionless systems, and to Ostriker (1999), who performed an equivalent treatment for a gaseous medium.

Dynamical friction arises when a massive perturber moving through some background (stellar or gaseous) medium creates in it a density wake. The wake trails the perturber and exerts on it gravitational force in the direction opposite to the motion, thus acting as a brake and earning this interaction a name: “dynamical friction." Consider, for example, an encounter of a MBH with mass M moving at a velocity \(\mathbf{v}\) relative to an isotropic background of stars with much lower mass, and a total mass density \(\rho \). If the stars have a three-dimensional Maxwellian velocity dispersion \(\sigma \), the drag experienced by the MBH is (e.g., Binney and Tremaine 1987)

$$\begin{aligned} {d\mathbf{v}\over {dt}}=-{4\pi \ln \varLambda \, G^2 M\rho \over {v^3}} \left[ \mathrm{erf}(X)-{2X\over {\sqrt{\pi }}}e^{-X^2}\right] \mathbf{v} \end{aligned}$$

where \(X\equiv v/(\sqrt{2}\sigma )\), \(\ln \varLambda = \ln (b_{\mathrm{max}}/b_{\mathrm{min}}) \sim 10-20\) is the Coulomb logarithm, and \(b_{\mathrm{min}}\) and \(b_{\mathrm{max}}\) are the smallest and largest effective impact parameter, respectively. The characteristic timescale for the MBH to decelerate due to multiple encounters with stars is then

$$\begin{aligned} t_{\mathrm{df}} = v/(dv/dt)\sim v^3/(4\pi G^2 M \rho )\; . \end{aligned}$$

An important implication of this expression is that MBHs of larger mass, immersed in denser environments, can sink faster toward the center of the remnant galaxy.

The exact timescale for decay of the MBH’s orbit in the remnant stellar bulge will depend on the properties of the stellar bulge, such as the distribution of stars and the stellar velocity dispersion \(\sigma \), which provides a characteristic speed of stars relative to the MBH. For example, following Antonini and Merritt (2012, see their Eq. 19), who calculate the effect of stellar dynamical friction in nuclear star clusters around MBHs, we estimate the inspiral time for a \(10^6\,M_\odot \) MBH initially located at \(r=100\,\)pc from the cluster center

$$\begin{aligned} t_{\mathrm{df}} \approx 45\,\mathrm{Myr} \, \left( \frac{r}{100\,\mathrm{pc}} \right) ^2 \left( \frac{\sigma }{100\,\mathrm{km\,s^{-1}}} \right) \left( \frac{M}{10^6\,M_\odot } \right) ^{-1} \left( \frac{\ln \varLambda }{15} \right) ^{-1} \,. \end{aligned}$$

Note that in the fiducial example considered in Eq. (4) inspiral from 1 kpc due to stellar dynamical friction takes 4.5 Gyr and is shorter for more massive black holes. In the estimate above, the stellar background is described by the mass density profile \(\rho (r) =\rho _0(r/1\,\mathrm{pc})^{-1.8}\), where \(\rho _0 = 1.5\times 10^5\,M_\odot \,\mathrm{pc^{-3}}\) is the normalization. This profile is an extrapolation of the observed distribution of stars in the central 10 pc of the Galactic Center (e.g., Oh et al. 2009). Stellar density profiles that decrease less rapidly with radius would result in even shorter timescales. See for example an estimate in Eq. (5) in Souza Lima et al. (2017), who instead assume a stellar distribution consistent with \(\rho \propto r^{-1}\) (i.e., the Hernquist bulge; Hernquist 1990).

If the remnant galaxy contains gas, the gas can also play an important role for pairing of MBHs. For example, Ostriker (1999) evaluated the gravitational drag exerted by a uniform-density gas background on a massive perturber on a linear trajectory.Footnote 8 Figure 10, adapted from this work, shows a comparison of the magnitude of the dynamical friction force, exerted by the gas with the same density as a purely collisionless background, and with a sound speed \(c_s\), equal to the velocity dispersion \(\sigma \) of the collisionless system. Ostriker (1999) found the following results as a function of the Mach number, \({{\mathscr {M}}}\equiv v/c_s\), where v is the speed of the massive object relative to the local standard of rest of the gas:

  1. 1.

    For \({{\mathscr {M}}}\gg 1\), the gas drag is the same as the drag against a collisionless background, for the same average density \(\rho \) and a velocity dispersion \(\sigma \) of the collisionless background equal to \(c_s\).

  2. 2.

    For \({{\mathscr {M}}}< 1\), the drag force is larger in a collisionless medium than in a gaseous medium. This is because the gas pressure forces result in a more symmetric distribution around the massive object.

  3. 3.

    For \({{\mathscr {M}}}\sim 1\), the dynamical friction force due to the gas can exceed the collisionless drag by a factor of four.

    Fig. 10
    figure 10

    Image(s) reproduced with permission from Ostriker (1999), copyright by AAS

    Magnitude of the dynamical friction force, as a function of the Mach number, \(\mathscr {M} = v/c_s\). The force is exerted either by the gas (solid lines) or by the collisionless background (dashed) of the same density, and with a sound speed of the gas equal to the velocity dispersion of the collisionless system. From top to bottom, the lines correspond to the dynamical friction force contributed by regions of increasing size around the perturber.

3.2.2 Implications for formation and properties of gravitationally bound MBHs

Because of its efficiency relative to the collisionless case, gaseous drag has been extensively investigated in simulations that follow pairing of MBHs in gas-rich mergers of galaxies (see Mayer 2013, 2017, for reviews). When pairing is successful, gaseous dynamical friction is capable of transporting the MBHs from galactic radii of few hundred pc to \(\sim 1\) pc on timescales as short as \(\sim 10^7\) yr (for details see, e.g., Escala et al. 2004, 2005; Dotti et al. 2006; Mayer et al. 2007; Van Wassenhove et al. 2014), and in many cases, tens of times faster than stellar dynamical friction (e.g., Berczik et al. 2006; Preto et al. 2011; Khan et al. 2011; Holley-Bockelmann and Khan 2015; Vasiliev et al. 2015; Pfister et al. 2017). The propensity of gas to effectively couple with the orbiting MBHs can be understood by recalling that unlike collections of stars, gas can radiate energy. Thus, again unlike stars, the same parcels of gas can interact multiple times with a black hole pair. As a result, in the extreme case that only gas is present, the binary can in principle be induced to spiral inwards without severely depleting the gas reservoir.

Simulations of isolated galaxy mergers, however, show that MBH pairs with mass ratios \(q< 0.1\) are less likely to form gravitationally bound binaries within a Hubble time at any redshift, primarily due to the inefficiency of dynamical friction on the lower-mass MBH (Callegari et al. 2009, 2011). They find that in moderately gas rich galaxies MBH pairs with initially unequal masses tend to evolve toward equal masses, through preferential accretion onto a smaller MBH. This trend is also consistent with that found in cosmological simulations Illustris (Vogelsberger et al. 2014; Kelley et al. 2017), ROMULUS25 (Tremmel et al. 2018; Ricarte et al. 2021), and a suite of cosmological zoom-in simulations (Bellovary et al. 2019, 2021); moreover, evidence for wandering MBHs in nearby dwarf galaxies has recently been found (Reines et al. 2020). These works show that low-mass galaxies, which lack a dense stellar core, are more likely to become tidally disrupted early in the process of merging. As a consequence, they deposit their MBHs at large radii, resulting in a population of ‘wandering’ MBHs. If so, this signals a preference for \(q> 0.1\) mergers of MBHs. In the circumstance that the opposite is suggested by observations of GW mergers, this would point to an existence of a mechanism other than dynamical friction that can promote pairing of MBHs of disparate masses.

Another important consideration for the pairing timescale of MBHs is the overall rotation of the stars and gas in the merged galaxy. For example, if a dominant component of overall speed of stars is in bulk motion, such as rotation, the velocity dispersion of the stars is then less than if their motion were random. Equation (2) indicates that the impact of dynamical friction (and more generally, dynamical relaxation) is stronger in systems with lower velocity dispersion \(\sigma \). Studies of rotating systems confirm their fast evolution (Einsel and Spurzem 1999; Kim et al. 2002, 2004; Berentzen et al. 2009; Amaro-Seoane et al. 2010a). Consequently, if the collision of galaxies that leads to the formation of dual MBHs also imparts significant rotation, as is likely in merger remnants, than the orbital evolution time of MBHs is much shorter than in non-rotating systems. Holley-Bockelmann and Khan (2015) for example find that for all other things being the same, rotating stellar bulges drive MBHs more efficiently through the dynamical friction and three-body scattering phase, resulting in timescales that are between 3 (for corotating systems) and 30 times shorter (for counterrotating systems) than the non-rotating bulges.

The latter timescale, obtained for systems in which a majority of stars are rotating in the opposite sense from the MBH pair, stems from the ability of counterrotating stellar orbits to effectively sap the orbital angular momentum of the MBHs. Therefore, the same effect should produce high eccentricity MBH pairs and bound binaries. This is important because if the eccentricity can truly increase arbitrarily close to unity, then the MBHB can get to a point where gravitational radiation is important and it will merge. This expectation is confirmed by studies that follow the evolution of binary eccentricity with respect to the host galaxy and find low eccentricities (\(e\lesssim 0.1\)) when the MBH binary and the galaxy are corotating, and high eccentricities (\(e\lesssim 1\)) and nearly radial orbits in the case of counterrotation (Amaro-Seoane et al. 2010a; Sesana et al. 2011; Gualandris et al. 2012; Wang et al. 2014; Holley-Bockelmann and Khan 2015; Mirza et al. 2017; Rasskazov and Merritt 2017). Furthermore, these studies find that MBH pairs and binaries, whose orbital angular momentum is initially misaligned with respect to that of the stellar cusp, tend to realign their orbital planes with the angular momentum of the cusp. Similar outcomes have been observed for MBHs evolving in gaseous circumnuclear disks. Specifically, the corotation of MBHs with the disk results in efficient circularization of the black holes’ orbits (Dotti et al. 2007; Li et al. 2020a). In the counterrotating case, however, gaseous dynamical friction drives the increase in MBH eccentricities until they experience “orbital angular momentum flip" and start to corotate with the disk before a MBHB forms (Dotti et al. 2009b).

Simulations of isolated galaxy mergers have also revealed that pairing of MBHs may be inefficient in clumpy galaxies, such as the massive star-forming galaxies detected at redshifts \(z\sim 1-3\). This is because gravitational scattering of MBHs by giant molecular clouds and stellar clusters can cause them to exhibit a random walk, rather than a smooth inspiral. Consequently, the orbital evolution of MBHs in such environments can be substantially slower (by one or two orders of magnitude relative to the timescale quoted above), or can stall permanently in some cases, at distances \(\sim 10\)\(10^3\,\mathrm{pc}\) from the center of the remnant galaxy (Fiacconi et al. 2013; Roškar et al. 2015; Tamburello et al. 2017). The long MBH pairing timescales obtained in these works leave room for the remnant galaxy to experience multiple mergers in this period of time. If so, the existence of triple, or even quadruple, MBHs in a single merger galaxy may be common, highlighting the need to understand their frequency of occurrence, and importance of dynamical interactions of multi-MBH systems. For example, using the Galaxy Zoo project catalog, Darg et al. (2011) find that a fraction of galaxy mergers involving more than two galaxies at a time is at least 2 orders of magnitude less than that of mergers of galaxy pairs. Liu et al. (2011) provide the incidence of multiple MBH in galaxy mergers, by investigating systems in which constituent MBHs accrete and shine as AGNs. They report the fraction of kpc scale AGNs pairs as \(\gtrsim 3\times 10^{-3}\), and kpc scale AGN triples as \(\gtrsim 5\times 10^{-5}\), both among optically selected AGNs at low redshift.

Interestingly, gravitational interaction of the MBH with the surrounding gas is quite local. For example, for a black hole with mass \(\sim 10^6\,M_\odot \) most of the gravitational drag force is contributed by the gas that resides within only a few parsecs from the MBH (Chapon et al. 2013). This proximity implies that the gas dynamical friction wake can be strongly affected, and possibly obliterated, by irradiation and feedback from an accreting MBH. Indeed, studies of dynamical evolution of MBHs find that dynamical friction can be significantly reduced due to the “wake evacuation" caused by purely thermal feedback from MBHs in simulations that follow gravitationally recoiled MBHs (see Fig. 11; Sijacki et al. 2011) and black hole pairing (Souza Lima et al. 2017; del Valle and Volonteri 2018). These results bring into question the assumed efficacy of gaseous dynamical friction. In this context, Park and Bogdanović (2017), Toyouchi et al. (2020) and Gruzinov et al. (2020) investigated the efficiency of gaseous dynamical friction in the presence of radiative feedback. The first two groups used local radiation hydrodynamic simulations and found that ionizing radiation, which emerges from the innermost parts of the MBH’s accretion flow, strongly affects the dynamical friction wake and renders gas dynamical friction inefficient for a range of MBH masses and gas densities. They find that MBHs in this regime tend to experience positive net acceleration, meaning that they speed up, contrary to the expectations for gaseous dynamical friction in the absence of radiative feedback. This effect is more severe at the lower mass end of the MBH spectrum which, compounded with inefficiency of the gas drag for lower mass objects in general, suggests that \(\lesssim 10^7\,M_\odot \) MBHs have fewer means to reach the centers of merged galaxies and form bound pairs with their counterpart MBHs. Indeed, Li et al. (2020b, 2022) show a significant suppression of pairing probability of such MBHs in merger galaxies with gas, in the presence of radiative feedback. This is of importance because systems of MBHs in this mass range are direct progenitors of merging binaries targeted by the space-based GW observatory LISA.

Fig. 11
figure 11

Image(s) reproduced with permission from Sijacki et al. (2011), copyright by the authors

Mass-weighted gas density maps from hydrodynamic simulations of recoiling MBHs in massive gas-rich galaxies. Left (right) panel shows the results of the simulation without (with) feedback exerted by the accreting MBH. The scenario with feedback results in the evacuation of gas from the wake following the MBH. Both panels show the location of the MBHs approximately 40 Myr after the recoil and the dotted lines trace their orbit from the start of each simulation.

Another important effect for the gaseous media is that when the gas accreting onto a MBH has much greater angular momentum than the hole itself, the MBH tends to align its spin axis with the angular momentum of the gas. This effect was originally proposed by Bardeen and Petterson (1975), and several papers have investigated important aspects of this process (Papaloizou and Pringle 1983; Natarajan and Pringle 1998; Natarajan and Armitage 1999; Sorathia et al. 2013a, b; Tremaine and Davis 2014; Morales Teixeira et al. 2014; Zhuravlev et al. 2014; Nealon et al. 2015; Polko and McKinney 2017; Hawley and Krolik 2018; Liska et al. 2019). Given that the radius of the accretion disk around an individual MBH can be thousands of times, or more, greater than the radius of the black hole, it follows that the angular momentum of the disk can dominate if it has more than a few percent of the mass of the MBH. An important implication of this statement is that it is, in principle, easy for the accretion flow to reorient the spin of the MBH. Note that this is different than the amount of gas required to significantly change the spin magnitude of the MBH, as conservation of angular momentum in this case requires accretion of the amount of gas comparable the original black-hole mass.

A major reason that this matters astrophysically is that, as we discuss in Sect. 3.4, mergers between holes with coaligned spins lead to moderate GW kicks, whereas mergers between comparable-mass black holes with randomly oriented spins can produce kicks of thousands of kilometers per second, sufficient to eject the remnant from any galaxy. This led Bogdanović et al. (2007) to propose that gas alignment was effective in reorienting MBH prior to their becoming a gravitationally bound binary. Calculations by Miller and Krolik (2013) further find that gas torques are effective in aligning the MBH spins and their orbit with the large scale gas flow not only at large orbital separations but also well into the final parsec, where the two form a gravitationally bound binary. Simulations by Dotti et al. (2009b, 2010) and Perego et al. (2009) support the spin-alignment hypothesis, with the refinement that because the large gas disk in the inner few hundred parsecs of a galaxy will be somewhat turbulent, accretion is not always in the same direction and hence, rather than being exactly aligned, the spins of MBHs will be mostly aligned. This effect is illustrated in Fig. 12, which shows the efficiency of alignment in circumnuclear disks that correspond to more turbulent (hot) or less turbulent (cold) gas flows. This level of alignment reduces the probability of a large kick but does not eliminate it, and depending on specific assumptions one every few thousand or few tens of thousands of galaxy mergers might result in a kick exceeding 2,000 km \(\hbox {s}^{-1}\) (Lousto et al. 2012).

Fig. 12
figure 12

Image(s) reproduced with permission from Dotti et al. (2010), copyright by the authors

Density distribution of the initial (\(\theta _i\)) and final (\(\theta _f\)) angles of MBH spins with the vector of the binary orbital angular momentum. Left (right) panels refer to MBHs embedded in cold (hot) circumnuclear gas disks, described with a polytropic equation of state with \(\gamma = 7/5\) (\(\gamma = 5/3\)). Upper (lower) panels refer to the spin of the secondary (primary) MBH, both with mass \(M=4\times 10^6\,M_\odot \). Dark, medium and light-grey surfaces refer to high density regions encompassing 68.3, 95.5 and 99.7 per cent of the realizations. \(\theta _i\) has been sampled isotropically, and the magnitudes of the dimensionless spin parameters (not shown) have been sampled from a constant probability distribution, between 0 and 1.

It is worth noting that there are also works proposing that the packets of gas that arrive at MBHs are insufficient to reorient their spin axes. In this picture, the black holes accrete small quantities of gas stochastically (as opposed to from a disk with a well defined angular momentum) and as a consequence, the majority of MBHs are expected to spin slowly (King et al. 2005; King and Pringle 2006, 2007; King et al. 2008; Wang et al. 2009; Nayakshin et al. 2012; Lodato and Gerosa 2013). The stochastic accretion hypothesis so far appears in tension with the spins of MBHs inferred from the X-ray observations of AGNs with relativistically broadened Fe K\(\alpha \) lines (Iwasawa et al. 1996; Fabian et al. 2002; Reynolds and Nowak 2003; Brenneman and Reynolds 2006, 2009; Schmoll et al. 2009; Brenneman et al. 2011; Reynolds et al. 2012). If the properties of these observed MBHs are representative of the broader population in the Universe, these measurements imply that the dimensionless spin magnitudes of most MBHs exceed \(s=0.6\). If not by the EM measurements, this hypothesis will ultimately be tested by the LISA GW observatory, which is expected to measure the distribution of MBH spins with exquisite precision.

3.3 Effects after MBH are gravitationally bound to each other

If MBH inspiral due to the stellar and gas dynamical interactions described in previous section is efficient, it eventually results in formation of a gravitationally bound binary. Qualitatively, a gravitationally bound MBHB forms when the amount of gas and stars enclosed within its orbit becomes comparable to its mass, M. This orbital separation is comparable to the radius of gravitational influence of a single MBH, where the circular velocity around the black hole equals the stellar velocity dispersion, \(r_{\mathrm{inf}} \approx GM/\sigma ^2\). Combining this with the \(M-\sigma \) relationship in Eq. (1) (McConnell and Ma 2013) gives

$$\begin{aligned} r_{\mathrm{inf}} \approx 3.2\,\mathrm{pc}\,\, M_7^{0.645} = 6.6\times 10^{6} r_g \,\, M_7^{-0.355}\; . \end{aligned}$$

where \(M_7 = M/10^7M_\odot \) and \(r_g = GM/c^2\approx 0.1~\mathrm{au}~(M/10^7~M_\odot )\) is the gravitational radius. Therefore, bound binaries form at separations of \(\sim 1-10\) pc for for a wide range of MBHB masses and host galaxy properties.

3.3.1 Three-body scattering of stars and interactions with additional MBHs

The simplest picture one can consider for a MBHB in this stage is that of a gravitationally bound binary that evolves due to interactions with individual stars in a completely spherically symmetric potential. For stars that come close enough to the binary, three-body interactions lead to ejection of stars with greater energy than they had initially, a process that leads to gradual hardening of the MBHB. In the limit where the MBHB masses are much larger than the masses of the interacting stars, the work of Quinlan (1996) is especially relevant. He showed that, given an unlimited supply of stars, the binary would harden at a rate given by the dimensionless parameter

$$\begin{aligned} H={\sigma \over {G\rho }}{d\over {dt}}\left( 1\over a\right) \,, \end{aligned}$$

where he found \(H\approx 10-20\) in the cases most relevant to us, in which the stellar speeds at infinity are much less than the binary orbital speed. In this case, the parameter H provides a dimensionless characterization of the rate of change of the binding energy of the binary. The associated hardening timescale can then be estimated as

$$\begin{aligned} t_{\mathrm{h}} = \left| \frac{a}{{\dot{a}}} \right| = \frac{\sigma }{G\rho aH} = 1.5\times 10^6\,{\mathrm{yr}}\,\left( \frac{\sigma }{100\,\mathrm{km\,s^{-1}}} \right) \left( \frac{10^4\,M_\odot \,\mathrm{pc^{-3}}}{\rho } \right) \left( \frac{0.1\,\mathrm{pc}}{a} \right) \left( \frac{15}{H} \right) \,, \end{aligned}$$

and it corresponds to the time that a MBHB of mass \(\sim 10^7\,M_\odot \) spends at orbital separation of \(a = 0.1\,\)pc in a nuclear stellar cluster with velocity dispersion of \(\sigma = 100\,\mathrm{km\,s^{-1}}\) and stellar mass density of \(\rho = 10^4\,M_\odot \,\mathrm{pc^{-3}}\). Note also that the MBHB spends increasingly longer times at smaller orbital separations and thus, its evolution slows down progressively.

The implication of Eq. (7) is that three-body interactions with stars can in principle lead to very efficient orbital evolution of MBHBs, with a wide range of masses, given an unlimited supply of stars. This assumption is reasonable in the stages of evolution when the MBHB orbital speed is comparable to the velocity dispersion of the stars, because stars are not completely ejected and can return for another interaction. When the MBHB orbital speed is significantly greater than \(\sigma \) however, interacting stars are ejected. In that scenario, the supply of stars that can interact with the MBHB is depleted and must, ultimately, be replenished if the binary is to continue its evolution. The timescale on which the stars are replenished therefore determines the rate of the binary orbital evolution.

This timescale is ordinarily determined by gravitational two-body interactions between stars in the nuclear star cluster around the MBHB, which can change the energy of a star by the order of itself. This is referred to as the energy relaxation time (Binney and Tremaine 1987; Spitzer 1987) and is directly related to the dynamical friction time shown in Eq. (4), as it describes the same type of gravitational interaction between the two point masses. For example, using Eq. (4) one can estimate a relaxation time of about 41 Gyr for a \(M=1\,M_\odot \) star residing in the nuclear star cluster with \(\sigma = 100\,\mathrm{km\,s^{-1}}\), from an initial radius \(r \approx 3\,\)pc, corresponding to the influence radius of a \(10^7\,M_\odot \) binary. Such a MBHB would thus be unable to shrink its orbit much below the influence radius by scattering of stars and would in principle have to rely on other processes, like emission of GWs, to continue its evolution. At this point, however, gravitational radiation driven inspiral would also take longer than a Hubble time for such a binary, leading to a hang-up in the evolution of the MBHB. This phenomenon, first described by Begelman et al. (1980), has been dubbed the “final parsec problem” (Milosavljević and Merritt 2003).

Thus, an important question for MBHBs evolving due to three-body interactions with stars is: are there physical scenarios in which the stars are supplied at a sufficiently high rate, so as to guarantee the delivery of the MBHBs to the GW regime and coalescence in less than a Hubble time? The answer depends on the orbital properties of the stars. If the interacting star is on an orbit with eccentricity e, then its specific angular momentum is \(\ell =\sqrt{GMa(1-e^2)}\), where M is the mass of the central object (either a single MBH or a MBHB). Two-body relaxation is a diffusive process, so the time required for the angular momentum to change by the order of itself is \((1-e^2)\) times the energy relaxation time. Thus, orbits with high eccentricities can change their angular momenta rapidly. The rapidity with which stars can change their angular momenta leads to the distinction between full and empty loss cones (for discussion of stellar dynamics in the centers of galaxies, see, e.g., Peebles 1972; Frank and Rees 1976; Bahcall and Wolf 1976, 1977; Lightman and Shapiro 1977; Wang and Merritt 2004; Hopman and Alexander 2005).

In general, the “loss cone" is the solid angle inside of which a particular star can have an important interaction with a target object. For example, for tidal disruption around a single MBH, the loss cone at a given distance from the black hole is the set of angles such that if a star’s orbit is inside that solid angle, it will be disrupted. In the case of interactions with a MBHB, the loss cone is the solid angle such that the star will pass within \(\sim 2\) binary semimajor axes of the binary, where the star can have a significant interaction with it. If the angular momentum can change by of order itself or more in a single orbit, the orbit is in the full loss cone regime, so-called because any star that is ejected or destroyed from such an orbit will be replaced immediately by another star. In contrast, if the typical change in the angular momentum in one orbit is less than the magnitude of the angular momentum, then the orbit is in the empty loss cone regime, and loss of stars must be replaced diffusively, on the timescale discussed in the previous paragraph. In such scenarios, there is a decrease in the phase space density inside empty loss cones, leading to significant slow down or complete hang-up in the orbital evolution of the MBHB. These two regimes are illustrated in Fig. 9, which shows different evolutionary tracks for MBHBs that find themselves in full and empty loss cones.

Given the importance of the final parsec problem for plausibility of coalescences of MBHBs, there have been many papers devoted to its solution. One approach has been to consider the evolution of MBHBs in non-spherical potentials. The motivation for this has been twofold. Firstly, galaxies that are remnants of major mergers are not expected to be spherically symmetric. This is because the relaxation timescales in their nuclei (discussed above) are expected to exceed a Hubble time in many relevant physical scenarios. Secondly, if the underlying stellar potential is triaxial then, because it lacks the azimuthal or spherical symmetry which would guarantees conservation of angular momentum for individual orbits, individual stars can replenish the loss cone much more rapidly than would be possible via two-body relaxation. Indeed, for galaxies that can be described as triaxial ellipsoids, there are a substantial number of stars on “centrophilic" orbits that can plunge arbitrarily close to their center (Holley-Bockelmann et al. 2002; Merritt and Poon 2004). Studies of these orbits have demonstrated that even moderate triaxiality suffices to allow a MBHB to merge within a Hubble time (e.g., Holley-Bockelmann and Sigurdsson 2006; Berczik et al. 2006; Preto et al. 2011; Khan et al. 2011; Holley-Bockelmann and Khan 2015; Vasiliev et al. 2015; Pfister et al. 2017).

For example, recent studies show that in all cases, a MBHB forms and hardens on timescales of at most 100 Myr, coalescing on another few-hundred-megayear timescale, depending on the binary orbital eccentricity and the characteristic density of stars (Khan et al. 2018). Higher central densities provide a larger supply of stars that can efficiently extract energy from the MBHB orbit, and shrink it to the phase where GW emission becomes dominant, leading to the coalescence of the MBHs. Overall, the sinking of the MBHB in such studies takes no more than \(\sim 0.5\,\)Gyr after the merger of the two galaxies is complete, but this can in principle be much faster for plunging binary orbits.

In addition to non-spherical potentials, a number of studies have considered the evolution of MBHBs in rotating galactic nuclei (see the discussion in Sect. 3.2.2 and Amaro-Seoane et al. 2010a; Sesana et al. 2011; Gualandris et al. 2012; Wang et al. 2014; Holley-Bockelmann and Khan 2015; Mirza et al. 2017; Rasskazov and Merritt 2017). An important result of these studies is that in general, rotation has been found to increase the hardening rate of MBHB even more effectively than galaxy geometry alone. In this case too, the corotation (counterrotation) of a MBHB with stars in the galactic nucleus results in relatively low (high) binary orbital eccentricity. For MBHBs that end up on relatively eccentric orbits, a residual eccentricity may be preserved all the way into the GW regime, potentially allowing to distinguish different evolutionary scenarios from GW observations.

It is worth noting that the same types of interactions that lead to efficient binary hardening also result in some number of stellar tidal disruptions. As a result, the evolution of the tidal disruption rates happens in tandem with the binary hardening rates, since both are driven by the depletion of loss-cone orbits (Chen et al. 2011; Lezhnin and Vasiliev 2019). Specifically, the stellar tidal disruptions of unequal mass MBHBs can reach a rate \(\sim 10^{-1}\,\mathrm{yr^{-1}}\) (three orders of magnitude higher than the corresponding value for a single MBH) soon after the gravitationally bound binary is formed (Chen et al. 2009). Depletion of loss-cone orbits leads to a precipitous drop in the disruption rate after \(\sim 10^6\) yr, which asymptotes to the value typical of single MBHs (\(\sim 10^{-5}\) to \(\sim 10^{-4}\,\mathrm{yr^{-1}}\)) for binaries in the GW regime. While the MBHB hardening rate is only weakly influenced by tidal disruptions (Chen et al. 2011), they can nevertheless give rise to bright EM flares (Coughlin et al. 2017; Coughlin and Armitage 2018), that may allow identification of a galaxy that is the host to a gravitationally bound binary, as long as this scenario can be distinguished from a disruption by a single MBH and variability of regular AGNs.

While stellar dynamical processes seem capable of delivering many MBHBs into the GW regime, it is prudent to consider what happens to those that experience ineffective stellar scattering and stall as a result at sub-parsec orbital separations. Given enough time, the remnant galaxy could undergo subsequent mergers with other galaxies and acquire additional (i.e., third, fourth, etc.) MBHs in the process (Mikkola and Valtonen 1990; Heinämäki 2001; Blaes et al. 2002). As noted before, studies of galaxy pairs lead to the conclusion that, on average, all massive galaxies have experienced a merger event in the last 10 billion years (Bell et al. 2006; Lotz et al. 2008). Assuming an average binary lifetime of 1 billion years implies that about 10 percent of MBHBs may form a triplet. Indeed, Kelley et al. (2017) and Bonetti et al. (2018) find that about 20–30 percent of all MBHBs could involve the presence of a third MBH.

Some of the consequences of the intrusion of another MBH into the MBHB system are: (i) the enhanced rate of loss cone refilling due to perturbation of stellar orbits by the intruding MBH (Perets and Alexander 2008), (ii) chaotic, non-hierarchical three-body interactions, which tend to shrink the binary semimajor axis and to increase its orbital eccentricity (Valtonen and Mikkola 1991), and (iii) if they form a hierarchical triple, the merger time of the inner binary can be dramatically reduced due to eccentricity oscillations induced by the Kozai-Lidov mechanism (Blaes et al. 2002). The latter two effectively increase the binary eccentricity to values close to unity, resulting in the GW losses and eventual MBHB coalescence on a timescale of \(\sim {\mathrm{few}}\times 10^8\,\)yr. All of them are likely to accelerate the MBHB coalescence and increase the ejection rate of (typically less massive) MBHs (Hoffman and Loeb 2007; Amaro-Seoane et al. 2010b; Ryu et al. 2018; Bonetti et al. 2019). This bodes well for GW searches for MBHBs, as it provides multiple means for them to reach this evolutionary stage.

3.3.2 Interactions with the circumbinary disk

The evolution of sub-parsec MBHBs in predominantly gaseous environments has also been a topic of a number of theoretical studies in the past 15 years (Armitage and Natarajan 2005; MacFadyen and Milosavljević 2008; Bogdanović et al. 2008, 2011; Cuadra et al. 2009; Haiman et al. 2009; Hayasaki 2009; Roedig et al. 2012; Shi et al. 2012; Shi and Krolik 2015; Noble et al. 2012; Kocsis et al. 2012a, b; D’Orazio et al. 2013; Farris et al. 2014; Rafikov 2016; Schnittman et al. 2016; Tang et al. 2017; Miranda et al. 2017; Bowen et al. 2017, 2018, 2019; Tang et al. 2018; d’Ascoli et al. 2018; Moody et al. 2019, etc.). Most have focused on a particular physical scenario, commonly referred to as the binary in circumbinary disk, in which a MBHB is embedded within a geometrically thin and optically thick gas disk. This setup is of interest because a dense and relatively cold circumbinary disk can, in principle, act as an effective conveyor belt for angular momentum. Such disks are also thought to be radiatively efficient and similar to disks that power AGNs – thus, they can, as an added bonus, provide a luminous EM tracer for inspiraling MBHBs.

As noted in Sect. 3.2.2, simulations of isolated galaxy mergers indicate that MBH pairs with mass ratios considerably lower that \(q \approx 0.1\) are less likely to form gravitationally bound binaries within a Hubble time, primarily due to the inefficiency of dynamical friction on the lower-mass MBHs. If, at the point of formation of the gravitationally bound binary, the two MBHs have a mass ratio larger than a few percent, theoretical studies indicate that the binary torques can truncate a sufficiently cold circumbinary disk and create an inner low-density cavity with a radius corresponding to approximately twice the binary semimajor axis (see Lin and Papaloizou 1979; D’Orazio et al. 2016, and references above). MBHs in this phase can accrete by capturing gas from the inner rim of the circumbinary disk and can in this way maintain mini-disks bound to individual holes. The left panel of Fig. 13 illustrates the geometry of the circumbinary region for MBHBs in this evolutionary stage, found in simulations that have investigated this scenario.

Fig. 13
figure 13

Illustration of the geometry of circumbinary region after the binary with comparable mass MBHs has created a low density region in the disk. Left: Simulations indicate that MBHs in this evolutionary phase can accrete by capturing gas from the inner rim of the circumbinary disk and can maintain mini-disks bound to individual holes. Adapted from Bogdanović (2015). Right: Snapshot of gas surface density from a 2D hydrodynamic simulation of circumbinary disk around a MBHB with mass ratio 1:4. Orbital motion is in the counterclockwise direction. Superposed green arrows trace the fluid velocity. Adapted from Farris et al. (2014)

Simulations also indicate that despite strong binary torques, accretion into the central cavity continues unhindered relative to the single MBH case (e.g., D’Orazio et al. 2013; Farris et al. 2014; Shi and Krolik 2015). If so, this has two important implications: (1) that necessary conditions exist for efficient transport of angular momentum through the circumbinary disk and potentially, orbital evolution of the MBHBs (see the discussion below, however) and (2) by analogy with single MBHs, binary MBHs powered by accretion can also shine as AGNs. Another property that many simulations of MBHBs in circumbinary disks identify robustly is that in unequal-mass binaries, accretion occurs preferentially onto the smaller of the two objects, because it orbits closer to the inner edge of the circumbinary disk (e.g., Artymowicz and Lubow 1994; Günther and Kley 2002; Hayasaki et al. 2007; Roedig et al. 2011; Nixon et al. 2011; Farris et al. 2014). This indicates that the lower-mass secondary MBH may grow at a rate higher than the primary and that, given enough time, the binary may evolve toward an equal-mass configuration.

Thus, the simplest scenario that one can consider in this case is that of a MBHB orbit that decays under the influence of accretion torques. In this case the inner rim of the circumbinary disk follows the MBHB inward until the timescale for orbital decay by gravitational radiation becomes shorter than the viscous timescaleFootnote 9 of the disk (Milosavljević and Phinney 2005; Armitage and Natarajan 2005). At that point, the rapid loss of orbital energy and angular momentum through gravitational radiation can cause the binary to detach from the circumbinary disk and to accelerate towards coalescence. If the orbital evolution of the binary is driven by the circumbinary disk, the rate of shrinking of the MBHB orbit can be described in terms of the viscous inflow rate at the disk inner edge as

$$\begin{aligned} {\dot{a}}_{\mathrm{visc}}= & {} - \frac{3}{2}\,\alpha \left( \frac{h}{r} \right) ^2 v_{\mathrm{Kep}}\nonumber \\= & {} -1.1\times 10^{-8} c \left( \frac{\alpha }{0.1} \right) ^{4/5} \left( \frac{{\dot{M}}}{0.1{\dot{M}}_{\mathrm{E}}} \right) ^{2/5} \left( \frac{10^4 r_g}{a} \right) ^{2/5} \left( \frac{10^7 M_\odot }{M} \right) ^{1/5} , \end{aligned}$$

where \(\alpha \) is the dimensionless viscosity parameter and h/r is the geometric aspect ratio of the disk in the Shakura and Sunyaev (1973) accretion disk model. \(v_{\mathrm{Kep}} = (GM/2a)^{1/2}\) is the circular orbital speed of the circumbinary disk with the inner edge at \(r=2a\), around a central object, here assumed to be a binary with semimajor axis a and mass M. Here \({\dot{M}}_{\mathrm{E}} = L_{\mathrm{E}}/\eta c^2\) is the Eddington mass accretion rate, \(\eta \) is the radiative efficiency, \(L_{\mathrm{E}} = 4\pi GM m_p c/\sigma _{\mathrm{T}}\) is the Eddington luminosity, \(\sigma _{\mathrm{T}}\) is the Thomson cross section, and other constants have their usual meaning. We obtain the second line of Eq. (8) by using the expression for h/r for a gas pressure dominated disk (see Eq. 2.16 in Shakura and Sunyaev 1973).

The associated timescale for the inner edge of the disk to move inward is then

$$\begin{aligned} t_{\mathrm{visc}} = - \frac{a}{{\dot{a}}_{\mathrm{visc}}} \approx 1.7\times 10^6\,{\mathrm{yr}} \, \left( \frac{0.1}{\alpha }\right) ^{4/5} \left( \frac{0.1{\dot{M}}_{\mathrm{E}}}{{\dot{M}}} \right) ^{2/5} \left( \frac{a}{10^4 r_g} \right) ^{7/5} \left( \frac{M}{10^7 M_\odot } \right) ^{6/5} , \end{aligned}$$

where we assumed a binary with orbital separation \(a=10^4\,r_g\) surrounded by a disk with a steady state accretion rate of 10% of the Eddington limit. It is worth noting that the Shakura-Sunyaev solution for an accretion disk around a single MBH implies that for each M and \({\dot{M}}\) there is a critical radius beyond which the disk is gravitationally unstable to fragmentation and star formation. This truncation radius in general arises somewhere between \(10^2\,r_g\) and \(10^6\,r_g\), depending on the properties of the disk and the mass of the central object (see Eqs. 15 and 16 in Haiman et al. 2009, for example). This is commonly considered as the outer edge of the accretion disk, beyond which the transport of angular momentum transitions from accretion torques to some other mechanism, either stellar or gaseous. Given that the structure and stability of circumbinary accretion disks is an area of active research, the existence and location of such an outer truncation radius are still open questions.

Equation (9) implies that if accretion torques are responsible for evolution of the MBHB orbit, its semimajor axis should monotonically decrease with time until the binary reaches the GW regime. Since the MBHB-circumbinary-disk system has more angular momentum than the nominal Shakura-Sunyaev MBH-disk system, one may also expect the timescale for orbital evolution to be longer than estimated above by some factor that depends on the properties of the MBHB and circumbinary disk. The expectation of systematic shrinking of binary orbit has been confirmed by many theoretical works leading to 2017 (see the references above), based on hydrodynamic and magnetohydrodynamic simulations of MBHBs in circumbinary disks. Works after 2017, however, show that significant uncertainties in the MBHB migration still remain.

The basic challenge with determining whether and how fast the MBHB orbit shrinks arises from the fact that angular momentum loss by gravitational torques is nearly canceled by angular momentum gain in accretion (Shi et al. 2012), making quantitative determination of the net sensitive to details of the calculation (Dittmann and Ryan 2021). For example, using 2D hydrodynamic simulations Tang et al. (2017) found that the torques that are responsible for MBHB orbital evolution emerge from the gas very close to the MBHs, and are contributed by the small asymmetries in the shape of the gas distribution of the mini-disks. The implication of this result is that the timescale for orbital evolution is sensitive to both numerical effects (numerical resolution and dimensionality of the model) and numerical implementation of the physical processes (such as accretion, thermodynamic properties of the disk, etc.). Surprisingly, the simulations by Tang et al. (2017) showed that when MBHs are modeled as rapid sinksFootnote 10 and accrete gas on a timescale comparable to or shorter than the local viscous time, the binary experiences positive torques, leading to expansion of its orbit. A few other groups, which performed close studies of the angular momentum transfer to the MBHB and the resulting binary orbital evolution, also report positive orbital torques: (a) Miranda et al. (2017), based on 2D simulations of circular binaries with different mass ratios and eccentricities, (b) Muñoz et al. (2019) for equal-mass binaries with different eccentricities, modeled in 2D, and (c) Moody et al. (2019) based on their 2D and 3D simulations of circular binaries with equal-mass ratios, coplanar and misaligned with the circumbinary disk.

More recently, Tiede et al. (2020) modeled equal-mass MBHBs on circular orbits and found that binary orbital evolution switches from outspiralling for warm, thicker disks (with aspect ratios \(h/r \sim 0.1\)), to inspiralling for cooler, thinner disks at a critical value of \(h/r \sim 0.04\). They explain that the reason why a group of earlier studies find positive orbital torques is because they model disks with \(h/r = 0.1\). Dittmann and Ryan (2022) arrive to a similar conclusion and find an additional dependence of the orbital torque on the assumed value of kinematic viscosity. These results underscore that the evolution of MBHBs is sensitive to the viscous and thermodynamic properties of the disk (reflected in its temperature and h/r). They also raise additional questions regarding the degree to which the calculation of the MBHB orbital torques may be affected by other approximations (like the description of the accretion flow equation of state) or by effects not captured by most present-day simulations (such as loss of angular momentum due to magnetically or radiatively driven outflows). At the time of this writing, some insight is provided by a study by Williamson et al. (2022), the first one to investigate orbital evolution of binary AGNs at parsec-scale separations affected by radiation pressure from the AGNs themselves. The study shows that radiation pressure cyclically destroys MBH mini-disks, leading to reduced gravitational torques and accretion, and significantly increased coalescence time scale.

The uncertainties mentioned above should also affect the evolution of the MBHB orbital eccentricity, which is a function of both the orbital angular momentum and energy. The works mentioned above (Tang et al. 2017; Miranda et al. 2017; Muñoz et al. 2019; Moody et al. 2019; Tiede et al. 2020; Dittmann and Ryan 2021, 2022) assume a MBHB with fixed orbital elements, and thus make no predictions about the evolution of eccentricity. Earlier works that reported the inspiral of MBHBs found that in prograde binaries, the exchange of angular momentum with the circumbinary disk drives a steady increase in binary eccentricity (Armitage and Natarajan 2005; Cuadra et al. 2009), which saturates in the range of 0.45–0.6 (Roedig et al. 2011; Zrake et al. 2021). This finding seemed intuitive, as the comparable mass MBHBs were found to produce circumbinary disks with non-axisymmetric, lopsided cavities. These were in turn expected to induce the increase of binary orbital eccentricity through gravitational interactions. For retrograde MBHBs, the cancelation in angular momentum with the circumbinary disk was found to lead to growth in the orbital eccentricity, which is slow for initially circular binaries, and fairly rapid and leading to \(e\approx 1\) for initially eccentric binaries (Nixon et al. 2011; Roedig and Sesana 2014; Bankert et al. 2015; Schnittman and Krolik 2015; Amaro-Seoane et al. 2016). The question of MBHB orbital eccentricity is important for the final outcome of the binary evolution (stalling or coalescence), as very eccentric MBHBs that reach small enough pericentric distances may evolve directly into the GW regime and coalesce rapidly (see the next section). Similarly, the orbital eccentricity of the MBHB may also be imprinted in its EM and GW signatures at coalescence, as we discuss in Sect. 4.

As noted in Sect. 3.2.2, the MBHs are expected to achieve an effective alignment of their spin axes with the rotation axis of the circumnuclear disk (and thus, with each other) by the time they form a gravitationally bound binary (Bogdanović et al. 2007). This alignment is expected to be further enforced after they form a binary, if their evolution continues in a circumbinary disk with a well defined orbital angular momentum (Miller and Krolik 2013). Alternatively, there is no a priori reason to expect the alignment of the MBH spins or high spins in scenarios characterized by the absence of coherent accretion or absence of gas altogether (King et al. 2005; King and Pringle 2006, 2007; King et al. 2008; Wang et al. 2009; Nayakshin et al. 2012; Lodato and Gerosa 2013).

3.4 The gravitational-wave phase

When the binary separation is small enough, the evolution of the MBHB is driven primarily by gravitational radiation, rather than by interactions with stars or gas. The subsequent evolution and coalescence of the black holes is commonly divided into three approximately defined phases, illustrated in Fig. 14. The most protracted phase is inspiral, in which the black holes are sufficiently separated that their inward radial migration is slow compared to their orbital motion. The second phase is merger, in which the black holes plunge together and merge. The third phase is ringdown, in which the initially asymmetric combined horizon radiates away modes and settles into a Kerr configuration. The motivation for this classification originates from different techniques used to describe these evolutionary stages. Specifically, inspiral and ringdown phases can be treated semi-analytically and are modeled using the post-Newtonian technique and perturbation method, respectively. The merger phase, however, involves full nonlinear gravity and must usually be treated numerically (in some cases the Effective One Body approach can also be used to compute the full GW signal; Buonanno and Damour 1999, 2000; Damour et al. 2000).

Fig. 14
figure 14

Image(s) reproduced with permission from Baumgarte and Shapiro (2011), copyright by AIP

The loss of energy and angular momentum via the emission of gravitational radiation drives a binary to coalescence, which proceeds in three different phases, illustrated here in terms of the GW strain amplitude as a function of time.

Emitted gravitational radiation carries away some fraction of the MBHB energy and angular momentum, and it is illustrative to examine how this affects the properties of the holes in the three evolutionary stages mentioned above. Consider a binary with component masses \(m_1\) and \(m_2\), total mass \(M=m_1+m_2\), and reduced mass ratio \(\mu =m_1m_2/M\). Since the inspiral phase includes the evolution of the MBHB from infinity to the innermost stable circular orbitFootnote 11 (ISCO), the energy released is \(\mu \) times the specific binding energy at the ISCO, so that \(E_{\mathrm{inspiral}}\sim \mu \). The total energy emitted in the merger and ringdown phases may be estimated using the order of magnitude rule that the strain amplitude from a binary with semimajor axis a, at a luminosity distance \(d_L\) from us, is \(h\sim (G\mu /ac^2)(GM/d_L\, c^2)\). For the merger and ringdown phases, the semimajor axis is comparable to the size of the event horizon of the binary,Footnote 12\(a \sim M\), so \(h\sim \mu /d_L\). Since the GW luminosity is \(L\sim d_L^2\, h^2 f^2\), where f is the GW frequency, then \(L\sim \mu ^2 f^2\). If the merger and ringdown phase last a time \(\tau \), then the total energy released is \(E\sim \mu ^2 f^2\tau \). But the characteristic GW frequency at merger is \(f \sim 1/M\) and the characteristic time is \(\tau \sim M\), resulting in \(E_{\mathrm{merger}}\sim \mu ^2/M\), or a factor \(\sim \mu /M\) times the energy released in the inspiral. As a consequence, for a comparable-mass binary the energy emitted in the merger and ringdown could be a few percent of the rest mass energy of the binary, and of the same order as the energy emitted in the inspiral. For systems with extreme mass ratios however, the inspiral is likely to be the only detectable phase.

In most of the inspiral phase, down to the separation \(\sim 10-20\,M\), the pure gravitational radiation driven evolution is well described by quadrupolar radiation (Peters 1964). In this approximation, the orbital semimajor axis, a, and eccentricity, e, of the binary evolve as

$$\begin{aligned} \biggl \langle {da\over {dt}}\biggr \rangle =-{64\over 5}{G^3\mu M^2\over { c^5a^3(1-e^2)^{7/2}}}\left( 1+{73\over {24}}e^2+{37\over {96}}e^4\right) \end{aligned}$$


$$\begin{aligned} \biggl \langle {de\over {dt}}\biggr \rangle =-{304\over {15}}e{G^3\mu M^2\over { c^5a^4(1-e^2)^{5/2}}}\left( 1+{121\over {304}}e^2\right) \end{aligned}$$

where the angle brackets indicate an average over an orbit. One can show that these rates imply that the quantity

$$\begin{aligned} ae^{-12/19}(1-e^2)\left( 1+{121\over {304}}e^2\right) ^{-870/2299} \end{aligned}$$

is constant throughout the inspiral, meaning that for moderate to small eccentricities we have \(e\propto a^{19/12}\). These equations imply that the characteristic time to inspiral for high-eccentricity orbits is

$$\begin{aligned} t_{\mathrm{gw}}\approx 10^9~{\mathrm{yr}} \left( \frac{10^{18}~M_\odot ^3}{\mu M^2}\right) \left( \frac{a}{0.001~\mathrm{pc}}\right) ^4 (1-e^2)^{7/2}\; , \end{aligned}$$

and that the eccentricity also decreases on that timescale. The dependence of the characteristic inspiral time on e, as well as circularization due to emission of gravitational radiation can be understood by recalling that the GW luminosity depends very strongly on the orbital frequency (and thus speed) of two masses. For high eccentricities the orbital speed is significantly larger at pericenter than it is at most places in the orbit. If one simplifies by assuming that GWs are only emitted at pericenter (as in the impulse approximation), the loss of energy and angular momentum at pericenter means that the apocenter distance decreases with time, and consequently, so does the eccentricity.

The strong dependence of the inspiral time, \(t_{\mathrm{gw}}\propto a^4\), means that at large distances the binary inspirals slowly. This timescale should be compared to the characteristic timescale for the inner edge of the circumbinary disk to move inward, \(t_{\mathrm{visc}} \propto a^{7/5}\), given by Eq. (9). This implies that for MBHBs surrounded by circumbinary disks, the radial diffusion of the circumbinary disk initially keeps the inner edge of the disk at a radius \(r\approx 2a\) as the binary shrinks. When a becomes small enough however, the GW inspiral time can become shorter than the disk radial diffusion time, so that the MBHB may decouple from the disk (Pringle 1991; Artymowicz and Lubow 1994, 1996; Liu et al. 2003; Milosavljević and Phinney 2005). Comparison of the two timescales indicates that the radius of decoupling can be \(\sim 100\,M\) for comparable-mass binaries and accretion at rates comparable to Eddington. It is reduced if one member of the MBHB is much less massive than the other, because gravitational radiation is weaker in that case. The implied decoupling radius is however larger at lower accretion rates, as long as the disks are in the radiatively efficient regime (with luminosity above \(\sim 0.01\,L_{\mathrm{E}}\); see, e.g., Ichimaru 1977; Rees et al. 1982; Narayan and Yi 1994; Blandford and Begelman 1999). This is because the disks then have smaller vertical thickness and hence smaller inward radial speeds (see Eq. (8)).

The question of decoupling of the MBHB from the circumbinary gas disk before coalescence is important, because coalescences that occur in gas poor environments may not be accompanied by detectable EM counterparts. For that reason, this question has been examined in hydrodynamic simulations of inspiraling MBHBs in circumbinary disks. For example, Noble et al. (2012) pointed out that magnetic fields in the circumbinary disk are stretched as the binary shrinks, and this can lead to much larger stresses at its inner edge than previously realized. They find that the disk can follow the binary down to \(\sim 20\,M\), which has encouraging implications for the detectability of EM counterparts. Similarly, Tang et al. (2018) find that the MBHs capture gas from the circumbinary disk and continue to accrete efficiently via their own mini-disks, well after their inspiral outpaces the viscous evolution of the disk.

In addition to effects on the semimajor axis and eccentricity, Schnittman (2004) showed that the directions of the spins of the two black holes can change systematically during GW inspiral. In particular, if the more massive black hole is at least partially aligned with the orbital axis and the mass ratio is close to unity, then the spins of both holes tend to align with the orbital axis. This has been proposed by Kesden et al. (2010) and Berti et al. (2012) as an important mechanism to further align already partially aligned spins, which (as we discuss below) in turn reduces the kicks produced by the MBH merger. Table 1 of Kesden et al. (2010) shows that alignment is only effective if the mass ratio is very close to unity and that even for mass ratios of 1/3, alignment is ineffective. Thus, this alignment mechanism might play a role in a limited set of mergers. Interestingly, however, Gerosa et al. (2015a) show that even initially aligned spin configurations can be brought out of alignment as a consequence of an instability that exists for a wide range of spin magnitudes and mass ratios, and can occur in the strong-field regime near the merger. They show that when the spin of the higher-mass black hole is aligned with the orbital angular momentum and the spin of the lower-mass black hole is antialigned, binaries in these configurations are unstable to precession to large spin misalignment. If prevalent in real MBHBs, this instability provides a way to circumvent gas-driven astrophysical spin alignment that may be acquired at large binary separations (see discussion in Sects. 3.2.2 and 3.3.2), allowing significant spin precession prior to merger.

The orientation of spins, as well as the asymmetry in masses of the two black holes, play an important role for the resulting magnitude of the “GW kick”, received by the daughter black hole that forms in the coalescence. The merger phase is the origin of the majority of this kick, relative to the original center of mass of the binary. The small pitch angle of the orbit during MBHB inspiral means that net emission of momentum in GWs nearly self-cancels over an orbit. Analytical calculations of the kick during inspiral (Peres 1962; Bekenstein 1973; Fitchett 1983; Fitchett and Detweiler 1984; Redmount and Rees 1989; Kidder 1995; Favata et al. 2004; Blanchet et al. 2005; Damour and Gopakumar 2006; Kidder et al. 2007; Schnittman and Buonanno 2007; Schnittman et al. 2008) suggest that prior to the final orbit the net speed relative to the original center of mass is at most tens of km \(\hbox {s}^{-1}\). However, the momentum radiated in the last half-orbit before merger is not self-cancelled, coinciding with moments before coalescence when the emission of gravitational radiation is strongest.

The breakthroughs that ushered in stable numerical simulations of black hole coalescence through ringdown (Pretorius 2005; Campanelli et al. 2006; Baker et al. 2006) have allowed GW kicks to be computed for many configurations, from nonspinning black holes to fully generic mergers of comparable-mass black holes without special spin orientations. They showed that if the spins are aligned with each other (as opposed to antialigned or at some angle) and the orbital angular momentum, then the maximum kick is moderate: less than 200 km \(\hbox {s}^{-1}\) (González et al. 2007). However, starting in 2007 other configurations were found that can lead to much greater kicks. Campanelli et al. (2007a, b) showed that for equal-mass maximally spinning black holes with oppositely-directed spins in the orbital plane (configuration also known as the “super kick”), the kick can reach almost 4000 km \(\hbox {s}^{-1}\). A different, “hangup kick” configuration, in which the spins are partway between the orbital plane and orbital axis, was introduced in Lousto and Zlochower (2011) and Lousto et al. (2012). The maximum velocity predicted for this configuration was shown to be close to 5,000 km \(\hbox {s}^{-1}\).

As numerical simulations of black hole mergers are extremely computationally expensive, several groups have developed approximate analytic formulae, calibrated on simulations, that predict kick velocity as a function of mass ratio and spin magnitudes and directions (Baker et al. 2007, 2008; Lousto and Zlochower 2008, 2009; van Meter et al. 2010a; Lousto et al. 2010, 2012). We show the one by Lousto et al. (2012) that includes the super kick configurations

$$\begin{aligned} \mathbf{v}_{\mathrm{kick}}= & {} v_m\,{\hat{e}}_1 + v_\perp (\cos {\xi }\,{\hat{e}}_1 + \sin {\xi }\,{\hat{e}}_2) + v_\parallel \,{\hat{L}} \,, \end{aligned}$$
$$\begin{aligned} v_m= & {} A_m\, \frac{\eta ^2(1-q)}{1+q}(1+B_m\eta ) \,,\end{aligned}$$
$$\begin{aligned} v_\perp= & {} H\frac{\eta ^2}{1+q}(s_1^\parallel -qs_2^\parallel ) \,, \end{aligned}$$
$$\begin{aligned} v_\parallel= & {} \frac{16\eta ^2}{1+q}(V_{11}+V_AS_z+V_BS_z^2+V_CS_z^3) | \mathbf{s_1}^\perp - q\,\mathbf{s_2}^\perp | \cos {(\phi _\varDelta - \phi _1)} \,. \end{aligned}$$

The velocity component \(v_m\) refers to the kick velocity contribution from unequal masses and \(v_\perp \), \(v_\parallel \), are contributions from the spins. The indices \(\perp \) and \(\parallel \) refer to the components perpendicular and parallel to the orbital angular momentum, respectively. Here, \({\hat{e}}_1\), \({\hat{e}}_2\) are the orthogonal unit vectors in the orbital plane, \(\xi \) is the angle that determines the projection of the spin contribution to the kick velocity in the orbital plane (\(v_\perp \)) on \({\hat{e}}_1\) and \({\hat{e}}_2\), and \({\hat{L}}\) is the unit vector in the direction of the orbital angular momentum. \(s_1=(cS_1/Gm_1^2)\) is the dimensionless spin angular momentum of the more massive, primary black hole, and \(s_2\) is that of the secondary, with similar definition. The parameter \(q =m_2/m_1 \le 1\) is the mass ratio, \(\eta = q/(1+q)^2\) is the symmetric mass ratio, and \(S_z= 2(s_1^\parallel +q^2 s_2^\parallel )/(1+q)^2\). \(\phi _1\) is the phase angle associated with the primary black hole and \(\phi _\varDelta \) is the angle between the in-plane vector, defined in Lousto et al. (2012) as a linear combination of the two spins, and a fiducial direction at merger. The best fit coefficients are \(A_m=1.2\times 10^4\) km \(\hbox {s}^{-1}\), \(B_m=-0.93\), \(H=6.9\times 10^3\) km \(\hbox {s}^{-1}\), \(V_{11}=3677.76\) km \(\hbox {s}^{-1}\), \(V_A=2481.21\) km \(\hbox {s}^{-1}\), \(V_B=1792.45\) km \(\hbox {s}^{-1}\), \(V_C=1506.52\) km \(\hbox {s}^{-1}\), and \(\xi = 145^\circ \).

It is worth noting that improved, albeit more complex versions of the analytic formulae are now available in the literature (e.g., Healy and Lousto 2018). In addition to the analytic formulae, a family of non-analytic, so called “surrogate models”, has been developed more recently. Surrogate models are trained against numerical simulations and were shown to be almost indistinguishable from simulations in the portion of parameter space where \(s_1\) , \(s_2 \le 0.8\) and \(q \le 1/4\) (Varma et al. 2019a, b).

As discussed by Merritt et al. (2004) and illustrated in Fig. 15, GW kicks exceeding \(\sim 2,000\) km \(\hbox {s}^{-1}\) suffice to eject the remnant MBH from any galaxy. Kicks exceeding several hundred km \(\hbox {s}^{-1}\) are enough to eject remnant MBHs from galaxies comparable to or smaller than the Milky Way. Thus, if such kicks are produced in some galactic mergers, there may exist some number of “empty nest” galaxies without a central MBH or with the MBH offset from the galaxy center. We discuss possible observational signatures associated with this phenomenon further in Sect. 4.3.

Fig. 15
figure 15

Image(s) reproduced with permission from Merritt et al. (2004), copyright by AAS

Central escape speeds in units of \(\mathrm{km\,s^{-1}}\) from different types of stellar systems: core ellipticals (E\(\cap \)), power-law ellipticals (E\(\backslash \)), dwarf ellipticals (dE), dwarf spheroidals (dSph), and globular clusters (GC). The solid line is the mean escape speed from the DM halos associated with the luminous matter. The dashed line is the escape speed from the combined luminous+mean DM potentials for E galaxies.

4 Emission signatures before, during, and after coalescence

In this section we discuss the emission signatures that arise before, during and after coalescence. Because our goal is to review the EM counterparts to merging MBHs that are sources of GWs, we focus on stages when the binary evolution is driven by emission of gravitational radiation (inspiral, merger and ringdown) rather than interactions with stars or gas. As discussed in Sect. 3, this may happen when the MBHs are within few hundred to a thousand gravitational radii of each other, or \(\sim 10^{-3} (M/10^8\,M_\odot )\) pc, and hence their orbital period is a few years or less. At such separations, the EM signatures are most likely to be associated with gas accretion flows surrounding the MBHBs. As noted in previous section, stellar tidal disruptions are not likely to play an important role for MBHBs in the GW-dominated phase of their inspiral, since their enhanced rate would have long since asymptoted to the value typical of single MBHs. We therefore focus attention on the MBHBs in circumbinary disks and assess pathways to EM signatures by considering several questions:

  • What is the semimajor axis at which MBHB evolution driven by stellar and gas interactions transitions to binary evolution driven by gravitational radiation?

  • What is the orbital eccentricity of the MBHB once it transitions to GW-driven evolution? Because emission of GWs leads to circularization, this transition determines the eccentricity of the MBHB at smaller separations and thus affects possibilities for time-variability of its EM signatures.

  • What fraction of the matter in the circumbinary disk eventually accretes onto either of the MBHs? Even if this fraction is small, the high efficiency of energy production in the mini-disks around each BH could mean that they dominate the overall luminosity.

  • Down to what spatial scale can the circumbinary disk follow the GW-driven shrinking of the MBHB? This could have a large impact on the simultaneity of EM and GW emission during the merger, as well as the EM luminosity of any merger-related event.

  • What is the role of periodic or quasi-periodic modulation of EM emission, as opposed to secular variability? Nearly periodic modulation, if observed over enough cycles, will be more easily distinguished from background noise than a gradual increase in intensity. Another important question is whether the fractional variation in intensity for periodic MBHB sources will be large enough to be detected, especially when combined with emission from the host galaxy.

We will keep these questions in mind as we lay out analytic estimates and discuss possibilities for production of characteristic EM signatures from merging MBHBs presented in the literature.

EM counterparts of merging MBHBs can originate from

 • The circumbinary disk.

 • Mini-disks and accretion streams.

 • Magnetized jets.

 • If all else fails: emission from the host galaxy.

4.1 Electromagnetic radiation before the merger

4.1.1 Emission from the circumbinary disk

The possible locations of EM emission before the MBHB merger are the circumbinary disk, the mini-disks around the individual black holes, and the streams feeding the mini-disks (see Fig. 13). Let us focus first on the circumbinary disk, where we will present some simple analytic estimates to guide the intuition and compare them with the results from simulations, paying attention to situations where they differ. The analytic estimates are obtained under the assumption that the circumbinary disk luminosity is produced by gravitational energy and angular momentum transport rather than by perturbations induced by the binary torques. We will also assume that the mass accretion rate through the disk is constant, and that it does not change as the MBHB inspirals and the inner rim of the circumbinary disk shrinks. As the binary inspirals, the disk then becomes more luminous and its spectrum peaks at higher photon energies.

This picture is similar to the scenario envisioned by many researchers (Pringle 1991; Artymowicz and Lubow 1994, 1996; Liu et al. 2003; Milosavljević and Phinney 2005). They noted that the GW-driven inspiral time decreases more rapidly with semimajor axis (\(t_{\mathrm{gw}}\propto a^4\), Eq. (13)) than does the viscous inflow time (\(t_{\mathrm{visc}}\propto a^{7/5}\), Eq. (9)). Therefore, when the binary semimajor axis decreases sufficiently, the binary may decouple from the disk. As noted in Sect. 3.4, comparison of the two timescales indicates that the decoupling may happen when \(a \sim 100\,M\) for comparable-mass binaries and accretion rates comparable to Eddington. If so, this implies that the gradual refilling of the hole after MBHB merger could provide a distinct signature of the merger (as suggested by Milosavljević and Phinney 2005). It also implies that the circumbinary disk will not be very luminous. The reason is that if the inner edge of the circumbinary disk is at \(r_{\mathrm{in}} \approx 2a \sim 200\,M\) when the binary decouples, then the energy is released with only \(M/r_{\mathrm{in}} \sim 0.5\)% efficiency. In comparison, the radiative efficiency of any matter accreting all the way to ISCO of either MBH is about 10%, so the luminosity of the circumbinary disk is at least a factor of twenty less. Based on these simple arguments, the emission luminosity from the MBH mini-disks is expected to dominate that from the circumbinary disk in a steady-state accretion flow some time before merger.

These considerations also imply that when the MBHBs merge, the circumbinary disk may be left behind at some appreciable distance. If so, there may be little matter around the MBHB at the moment of coalescence and hence, few opportunities for direct interaction of that matter with the dynamically changing spacetime near the MBHs that could produce a luminous EM counterpart. This early concern was mitigated by simulations (Noble et al. 2012; Farris et al. 2015a; Bowen et al. 2017; Tang et al. 2018), which indicate that the inner rim of the circumbinary disk is able to follow the binary to radii of \(\sim 10\,M\). This is also true for the spinning MBHBs, as the spins do not strongly affect the bulk properties of the circumbinary disk (Lopez Armengol et al. 2021). For the circumbinary disk with an inner rim radius of \(\sim 20\,M\), the energy release could then be \(\sim \) 5% and substantial enough to produce an EM luminous signature. Similarly, the inspiral time of the gas is then much shorter than previously assumed, thus allowing the MBHs to accrete all the way to coalescence. This is illustrated in Fig. 16, where the two MBHs are fed by the narrow gas streams till coalescence.

Fig. 16
figure 16

Image(s) reproduced with permission from Tang et al. (2018), copyright by the authors

Surface density of the gas around an equal mass \(10^6\,M_\odot \) MBHB evolving from the initial separation of \(60\,M\) (left panel) to merger (right), on a timescale of about a week. Despite the presence of an asymmetric low density cavity, the two MBHs continue to accrete gas carried by the narrow gas streams all the way through coalescence. The blue and green (orange and red) colors indicate low (high) surface densities.

The prime difficulty with gravitationally-powered circumbinary disk emission as a signature of binary MBHB coalescence is that the gradual increase in luminosity and change in spectrum is smooth and thus, might not be easily distinguishable from other gradual changes that exist naturally in disks around single AGN. As a reminder, we expect at most a few MBHB mergers per year out to a redshift \(z\sim 1\), compared to the millions of AGNs that exist to that redshift. Thus without a contemporaneous GW signal and decent sky localization to guide a search, one needs to identify a signature that exists in fewer than one AGN per million. It is not clear that smooth changes in the luminosity and spectral properties of the circumbinary disk would provide such an opportunity (see however the discussion of the “notches” in the total spectrum contributed by the circumbinary disk, the streams, and the mini-disks at the end of Sect. 4.1.2).

To explore other possibilities related to circumbinary disks, we return to the prospect of torques exerted by the MBHB on the disk. Consider first a circular binary. The magnitude of the torque it exerts on the circumbinary disk is likely to be time-independent, once we factor out the slow inward motion of the binary and disk. The torque is not, however, axisymmetric because the binary potential is not. Thus, there could be azimuthal modulation of the emission that appears periodic, depending on the viewing direction of the observer. How strong this modulation is depends on multiple factors. For example, if the disk is sufficiently optically thick that the radiation diffusion time is longer than an orbital period, one would expect that visible modulation of the emission would be strongly suppressed. If in contrast the radiation diffusion time is much less than an orbital period, the modulation might be visible, but the mass of the disk that is torqued is decreased as a result.

We can quantify these considerations by imagining a disk with a fixed aspect ratio h/r and surface density \(\varSigma \). Let the inner radius of the disk be \(r_{\mathrm{in}}=2a\) and the area \(f\pi r_{\mathrm{in}}^2\), where \(f<1\) is a fraction of the disk experiencing strong torques. Then the optical depth of the disk from the midplane is \(\tau _{\mathrm{T}}=\varSigma \kappa _{\mathrm{T}}\), where \(\kappa _{\mathrm{T}}\) is the opacity to Thomson scattering. For a MBHB with a total mass M, the orbital period at the inner edge of the disk is \(P_{\mathrm{in}}=2\pi (r_{\mathrm{in}}^3/GM)^{1/2}\). The radiation diffusion time from the midplane is \(t_{\mathrm{diff}}=(h/r)\tau _{\mathrm{T}} (r_{\mathrm{in}}/c)\). The condition \(t_{\mathrm{diff}}<P_{\mathrm{in}}\) thus implies

$$\begin{aligned} \varSigma < 2\pi \left( \frac{h}{r}\,\kappa _{\mathrm{T}} \right) ^{-1} \left( \frac{r_{\mathrm{in}}c^2}{GM}\right) ^{1/2}\; . \end{aligned}$$

This corresponds to a quantitative value of the surface density

$$\begin{aligned} \varSigma < 2\times 10^3~\mathrm{g~cm}^{-2} \left( \frac{h/r}{0.1} \right) ^{-1} \left( \frac{\kappa _{\mathrm{T}}}{0.4~\mathrm{cm}^2~\mathrm{g}^{-1}} \right) ^{-1} \left( \frac{r_{\mathrm{in}}}{200\,M}\right) ^{1/2} \,, \end{aligned}$$

and total mass

$$\begin{aligned} m < f\pi r_{\mathrm{in}}^2\,\varSigma \approx 30\,M_\odot \,f \left( \frac{h/r}{0.1} \right) ^{-1} \left( \frac{\kappa _{\mathrm{T}}}{0.4~\mathrm{cm}^2~\mathrm{g}^{-1}} \right) ^{-1} \left( \frac{r_{\mathrm{in}}}{200\,M}\right) ^{5/2} \left( \frac{M}{10^8~M_\odot }\right) ^2 \,. \end{aligned}$$

Thus, the portion of the circumbinary disk experiencing strong torques is likely to be small compared to the mass of the MBHB. Note also that the torque luminosity is unlikely to exceed the emission luminosity of the circumbinary disk produced by gravitational energy and angular momentum transport. If it did, such energy injection would increase significantly the thickness of the disk, facilitating fast radial inflow of the gas toward the MBHB. With that in mind, for the purposes of this section we consider the torque luminosity limited by the Eddington luminosity of the flow, \(L_t \lesssim L_{\mathrm{E}} \approx 1.3 \times 10^{46}\,\mathrm{erg\,s}^{-1} (M/10^8\,M_\odot )\).

For a circular MBHB the modulation of luminosity emitted by the circumbinary disk is likely be small, because even if all the energy is released at a single orbital phase, the modulation would rely on Doppler boosting, which is of order \(v_{\mathrm{orb}}/c\sim 0.1\) for \(a=100\,M\). Greater modulation is possible if the MBHB has significant eccentricity, because then the total torque also has a time modulation. This implies that the viability of this mechanism as a signature relies on the eccentricity of the binary. Several studies have shown that when the binary evolution is driven by torques from the circumbinary gas disk, then the MBHB orbital eccentricity tends toward higher values (Artymowicz 1992; Armitage and Natarajan 2005; Cuadra et al. 2009). More specifically, binaries that start their evolution with low to moderate eccentricities reach a limiting value of \(e\approx 0.4-0.6\) (Roedig et al. 2011; Zrake et al. 2021). Once the binary’s evolution is dominated by gravitational radiation however, the eccentricity drops roughly as \(e\sim f_{\mathrm{orb}}^{-19/18}\) for small to moderate eccentricities (Peters 1964). Thus, to determine the eccentricity when MBHB torques are relevant in this phase, we need to estimate the orbital separation at which the binary’s evolution is driven by GWs, and the largest separation at which we might be able to identify periodic fluctuations.

We estimate the first radius by comparing the torque luminosity to the luminosity of gravitational radiation. The GW luminosity for a binary with total mass M, symmetric mass ratio \(\eta \), semimajor axis a, and eccentricity e is (Peters 1964)

$$\begin{aligned} L_{\mathrm{GW}}=-{32\over 5}{G^4\eta ^4M^5\over {c^5a^5(1-e^2)^{7/2}}} \left( 1+{73\over {24}}e^2+{37\over {96}}e^4\right) \; . \end{aligned}$$

Setting \(e_{\mathrm{GW}}=0.6\) in this stage of evolution, we rewrite this as

$$\begin{aligned} L_{\mathrm{GW}}=65\left( \frac{\eta }{0.25} \right) ^4 \left( \frac{GM}{ac^2}\right) ^5 \left( \frac{c^5}{G} \right) \; . \end{aligned}$$

Equating this to \((L_{\mathrm{t}}/L_{\mathrm{E}}) L_{\mathrm{E}}\) gives the transition semimajor axis

$$\begin{aligned} a_{\mathrm{GW}}\approx 370\,M \left( \frac{M}{10^8\,M_\odot } \right) ^{-1/5} \left( \frac{L_{\mathrm{t}}}{L_{\mathrm{E}}} \right) ^{-1/5}\; , \end{aligned}$$

where we assumed an equal mass binary with \(\eta = 0.25\).

When it comes to the consideration of the maximum separation at which the periodic modulation in EM luminosity of the circumbinary disk can be reliably detected, we note that \(\sim 10\) cycles must be seen to be confident in the periodic nature of the signal and rule out an AGN with unusual stochastic variability. Thus, the longest MBHB orbital period we consider for the purposes of this estimate is \(P\sim 1\) yr. The semimajor axis that satisfies \(2\pi \sqrt{a^3/GM}=1\) yr is

$$\begin{aligned} a_{{\mathrm{yr}}}\approx 460\,M \left( \frac{M}{10^8\,M_\odot } \right) ^{-2/3}\;. \end{aligned}$$

Equations (23) and (24) indicate that \(a_{{\mathrm{yr}}} \sim a_{\mathrm{GW}}\) when \(L_{\mathrm{t}}/L_{\mathrm{E}} \lesssim 1\). According to Eq. (12), \(e\propto a^{19/12}\) for MBHBs in the GW regime, and so the eccentricity at \(a_{\mathrm{yr}}\) can be estimated

$$\begin{aligned} e_{\mathrm{yr}}\approx e_{\mathrm{GW}} \left( \frac{a_{\mathrm{yr}}}{a_{\mathrm{GW}}} \right) ^{19/12}\approx 0.6 \left( \frac{M}{10^8~M_\odot } \right) ^{-133/180} \left( \frac{L_{\mathrm{t}}}{L_E} \right) ^{19/60}\; , \end{aligned}$$

or \(e_{\mathrm{yr}}\approx 0.6\) if the expression above would suggest a higher eccentricity. Thus, if the torque luminosity is close to Eddington, the MBHB eccentricity and associated EM variability could be substantial at a one year period. To estimate the length of time during which such variability could be sustained we determine the characteristic GW inspiral time from a one year orbital period by substituting an expression for \(a_{\mathrm{yr}}\) in Eq. (13) to obtain

$$\begin{aligned} t_{\mathrm{GW}} \approx 2\times 10^4\,{\mathrm{yr}}\, \left( \frac{M}{10^8~M_\odot } \right) ^{-5/3} \,, \end{aligned}$$

where we evaluated \(t_{\mathrm{GW}}\) for an equal-mass MBHB and \(e_{\mathrm{yr}} = 0.6\).

It is illustrative to consider how many variable MBHB systems could be visible on the sky at any given time. Suppose that there is an instrument that can survey the entire sky over a \(\sim 1\) year period with sensitivity corresponding to a bolometric flux of \(F=10^{-12}\,\mathrm{erg\,cm}^{-2}\mathrm{s}^{-1}\). Such an instrument would be able to see a source with a luminosity L to a luminosity distance

$$\begin{aligned} d_{\mathrm{L}}\approx 10\,\mathrm{Gpc}\, \left( \frac{L}{L_E}\right) ^{1/2} \left( \frac{M}{10^8~M_\odot } \right) ^{1/2} \left( \frac{F}{10^{-12}\,\mathrm{erg\,cm}^{-2}\mathrm{s}^{-1}}\right) ^{-1/2}\; . \end{aligned}$$

Thus, a \(10^8\,M_\odot \) MBHB with a circumbinary disk that emits at nearly the Eddington luminosity would be visible to a redshift of \(z\sim 1.5\). The number density of galaxies that harbor \(\sim 10^8~M_\odot \) black holes is approximately \(10^{-3}~{\mathrm{Mpc}}^{-3}\) (Marconi et al. 2004). Since the comoving volume at \(z\sim 1.5\) is \(\sim 150\) \(\hbox {Gpc}^3\), this implies \(\sim 10^8\) such galaxies in that volume. Some estimates suggest that tens of percent of galaxies have had at least one, and possibly a few major mergers since \(z\sim 1\) (Bell et al. 2006; Lotz et al. 2008; Maller et al. 2006; Hopkins et al. 2010). Taking into account that the light travel time from \(z\sim 1.5\) is about 10 Gyr, we calculate the approximate merger rate of \((10^8)(0.1)(1)/10^{10}\,{\mathrm{yr}}\sim 10^{-3}\) \(\hbox {yr}^{-1}\). We estimated in Eq. (26) that the maximum extent of time over which the EM variability could be sustained is \({\mathrm{few}}\times 10^4\,\)yr, implying that there are potentially few 10s of such systems that are visible during the circumbinary disk phase, if they radiate at the Eddington luminosity. If instead the luminosity is a percent of Eddington, one can show that such sources would be visible to a redshift \(z\sim 0.2\). As a result, their number drops to a few systems at a time that are active and have a period of a year.

It is worth noting that the estimate above applies to the more massive end of MBHB population, targeted by the PTAs. It is consistent with findings by Zrake et al. (2021), based on high-resolution hydrodynamic simulations, who predict that comparable-mass PTA binaries should be detected with \(e \approx 0.4\)–0.5. The LISA sources on the other hand undergo GW circularization and are likely to enter the LISA band with a measurable eccentricity of \(10^{-2}\)\(10^{-3}\). The mergers of lower mass MBHB systems, targeted by LISA, Tian-Qin and similar space-based observatories, will be more numerous but less EM luminous in absolute terms, even if they radiate at the Eddington luminosity. For sub-Eddington systems in this mass range, stars in the host galaxy may outshine the emission from the circumbinary disk, particularly in the rest-frame infrared and optical band. This indicates that such systems may be comparatively more difficult to identify, at least based solely on their circumbinary disk emission (see Sect. 2.3 for predicted detections rates of MBHBs exhibiting quasi-periodic EM variability).

4.1.2 Emission from the mini-disks and accretion streams

As discussed in the previous section, a possible limitation of circumbinary disk emission is that its luminosity is restricted by its low efficiency, because the inner rim of the disk will remain at a large radius compared to the ISCO (located at \(6\,M\) for non-spinning black holes) for most of the inspiral. We therefore turn our attention to emission from mini-disks that may exist around the individual MBHs. In these disks, the matter spirals all the way to the ISCO and has the opportunity to release \(\sim \)10% of its mass-energy. If, for example, the inner edge of the circumbinary disk is at \(\sim 1000\,M\), it releases \(\sim 0.1\)% of the mass-energy of the matter. The hundred times higher efficiency of the mini-disks implies that only 1% of the circumbinary disk gas needs to make its way to the mini-disks for their luminosity to be comparable to that of the circumbinary disk. If a significantly greater fraction of the circumbinary gas spills over, the mini-disks can dominate the luminosity and might provide better opportunities for the EM signatures.

Hence, in order to tackle the question of the luminosity of the MBH mini-disks, one needs to understand how much gas is channelled to them from the circumbinary disk. Early models that investigated this question assumed the circumbinary disk to be axisymmetric as well as being vertically averaged. In such models there can be no accretion onto the individual MBHs, and there is significant pileup of matter at the inner edge of the circumbinary disk (Armitage and Natarajan 2002; Lodato et al. 2009; Chang et al. 2010; Kocsis et al. 2012a, b; Rafikov 2012). This led to the proposal of signatures of MBHBs evolution that rely on this pileup. Simulations that subsequently relaxed the axisymmetry condition yielded qualitatively different results. They find in particular that the nonaxisymmetric and time-dependent gravitational accelerations tend to fling matter out of the system or cause accretion, rather than acting as a simple barrier (Baruteau et al. 2012; D’Orazio et al. 2013; Farris et al. 2014; Shi and Krolik 2015, and others). They established that despite strong binary torques, accretion into the central cavity continues unhindered and is comparable to the single MBH case. They also found that the portion of the stream that is flung by the MBHB toward the inner rim of the circumbinary disk produces a non-axisymmetric density enhancement at its inner edge, often called a “lump” (Noble et al. 2012; Shi et al. 2012; Farris et al. 2014; Gold et al. 2014a). An interesting consequence of this density distribution is that the lump quasi-periodically modulates the accretion flux into the central cavity and the mini-disks, even when the orbital eccentricity of the MBHB is modest. The relative amplitude of the lump (and the associated EM periodic signal) were however found to diminish with greater magnetization of the accretion flow and the decreasing MBH mass ratio, vanishing completely between \(0.2<q<0.5\) (Noble et al. 2021).

We can use simple considerations to explore what could happen to the matter that follows a path that allows it to be captured by either MBH. For example, a key question related to the observability of modulation is whether the inspiral time through the mini-disks is significantly less than, comparable to, or significantly greater than the MBHB orbital time. We will see in the following few paragraphs that this depends on the orbital separation of the MBHB as well as the thermodynamic properties of the disk. If the inspiral time is long, then mini-disks can provide effective buffering of the incoming modulations in the accretion flux driven by the lump. In this case the lump period can still be imprinted in the low-energy (optical and infrared) emission associated with the streams and the cavity wall, but would be absent in accretion rates of the two MBHs (Westernacher-Schneider et al. 2021). If the inspiral time is short, then major modulation is possible, because the feeding rate to the binary mini-disks (which is modulated at the orbital period of the binary and the lump) would be reflected in the accretion rate onto the black holes. We next estimate the inspiral time through a mini-disk to an individual MBH and compare it with the orbital period of the binary. Suppose that the MBHB has mass ratio q, a semimajor axis a and an eccentricity e. The maximum extent of either mini-disk is set by its Roche lobe at the pericenter of the binary orbit, since the mini-disk that expands beyond that radius gets truncated by tidal forces from the other MBH.

Using the formula of Eggleton (1983), the Roche lobe radius around the lower-mass black hole is

$$\begin{aligned} \frac{r_2}{a(1-e)}={0.49q^{2/3}\over {0.6q^{2/3}+\ln (1+q^{1/3})}}\; . \end{aligned}$$

For example, in the equal-mass case \(q=1\) and \(e=0.6\), \(r_1=r_2=r = 0.38a(1-e) \approx 0.15a\). In the standard disk solution of Shakura and Sunyaev (1973), the inspiral time of the gas is given by the viscous timescale \(t_{\mathrm{visc}}=2r/[3\alpha (h/r)^2(GM/2r)^{1/2}]\) (see Eq. (9)), where we accounted for the fact that the mass of a single MBH is M/2. Comparison with the MBHB orbital period, \(P_{\mathrm{orb}}=2\pi (a^3/GM)^{1/2}\), yields

$$\begin{aligned} \frac{t_{\mathrm{visc}}}{P_{\mathrm{orb}}}\approx \left( \frac{\alpha }{0.1} \right) ^{-1} \left( \frac{h/r}{0.3} \right) ^{-2}\; . \end{aligned}$$

It follows that, if the mini-disk has sufficient geometrical thickness, it is possible that the inspiral time will be shorter than the orbital time. According to the Shakura and Sunyaev (1973) solution, the disks are this geometrically thick only at high accretion rates and at small radii, where they are supported by radiation pressure gradients. For example, at the Eddington accretion rate, \(h/r>0.3\) only when \(r<23\,M\). Thus, if the accretion rate to the individual MBHs is large, and the holes are fairly close together at pericenter (so that \(a \lesssim 150\,M\)), a modulation in the accretion rate from the streams may appear as a modulation in the accretion rate onto the holes.

When the MBHs are at such close separations, additional modulation effects arise as a consequence of the relativistic dynamics and the shape of the gravitational potential between the two MBHs. One is the mass exchange between the MBH mini-disks, which happens because the potential between the two MBHs becomes shallower than in the Newtonian regime, causing the quasi-periodic “sloshing” of gas at \(\sim 2\)–3 times the MBHB orbital frequency (Bowen et al. 2017). The second effect arises when the radius of the Hill sphere (the sphere of gravitational dominance) of an individual MBH becomes comparable to the radius of its innermost stable circular orbit. In this case, the absence of stable orbits for the gas in the mini-disks precludes them from maintaining the inflow equilibrium. As a consequence, the mini-disk masses show significant quasi-periodic fluctuations with time (Gold et al. 2014b; Bowen et al. 2018, 2019), potentially providing another time-dependent feature in the MBHB’s EM emission.

Because the size of the Hill sphere and ISCO depend on the black hole mass ratio and spins, one would expect that the total mass of the mini-disks and thus, the brightness of their EM signatures depend on these parameters. This dependence was studied in a series of recent works based on GRMHD simulations of mini-disks associated with inspiraling MBHBs with \(a \lesssim 20\,M\). Combi et al. (2021) for example find that whether the MBHs are spinning or not, the mass and accretion rate of mini-disks maintain periodicities set by the beat frequency between the orbital frequencies of the MBHB and an overdense lump in the circumbinary disk, corresponding to approximately 0.72 of the MBHB orbital frequency, \(\varOmega _{\mathrm{B}}\). Gutiérrez et al. (2021) subsequently performed a post-processing analysis of the EM emission from simulations presented in Combi et al. (2021) and report that it also exhibits periodicity driven by the lump dynamics, albeit at different characteristic frequencies corresponding to the radial oscillations of the lump (\(\sim 0.2\,\varOmega _{\mathrm{B}}\)) and at twice the beat frequency (\(\sim 1.44\,\varOmega _{\mathrm{B}}\)). They also show that for spinning MBHs with dimensionless spins \(s_1 = s_2 = 0.6\) aligned with the orbital angular momentum of the binary, the mini-disks are more massive, and consequently \(\sim 3-5\) times brighter relative to the non-spinning binary configurations. Regardless of the spin properties however, mini-disks in MBHBs are characterized by 25-70% lower radiative efficiencies than “standard” disks around single MBHs, and are thus expected to be dimmer. In another study of inspiraling MBHBs with \(a < 20\,M\), Paschalidis et al. (2021) show that the exact timing of the mini-disks disappearance and EM dimming, as they become smaller than the ISCO, also depends on the MBH spins. They propose that in inspiraling MBHB systems where an early GW detection allows a prompt EM follow-up, the timing of the mini-disk fading can provide a new probe of the MBH spins, different from the GW measurement.

However, another possibility exists. The accretion streams, which fall nearly ballistically from the inner edge of the circumbinary disk, will hit the outer edges of the mini-disks, producing shocks and emission that might have a different spectrum from the disk spectra. Roedig et al. (2014) find that the hotspots created by shocks should radiate Wien spectra with temperatures \(\sim 100\,\)keV and that their cooling time is smaller than the MBHB orbital time. If so, modulation in the stream rates may be directly reflected in modulation of the shock properties. The total energy release in such shocks might not be large however. Instead of releasing the \(\sim 10\)% of the mass-energy available at the ISCO, the impacts typically release only the binding energy at the Roche radius. In our equal-mass MBHB example, this amounts to the binding energy at \(r=0.15a\) around a mass M/2 black hole, so the specific energy release is only a few times higher than the specific energy released in the circumbinary disk (see Sect. 4.1.1). This may render this signature challenging to detect, given the current limitations of hard X-ray detectors.

Given the anticipated properties of emission from the circumbinary disk, the mini-disks and accretion streams, it is interesting to examine their relative contribution to the spectral energy distribution of the MBHB. A number of works in the literature considered the thermal spectrum from the components of this gas flow and the imprint in it created by the low density gap (Gültekin and Miller 2012; Kocsis et al. 2012a; Tanaka et al. 2012; Tanaka and Haiman 2013; Roedig et al. 2014; Farris et al. 2015b; Ryan and MacFadyen 2017; Tang et al. 2018). The left panel of Fig. 17 shows two examples of a thermal luminosity spectrum from Tanaka et al. (2012), calculated using a model of a thin, viscous disk for a \(10^9\,M_\odot \) MBHB and mass accretion rate \({\dot{M}} = 3\, {\dot{M}}_E\) through the circumbinary disk. These spectra include contributions from the circumbinary disk and the mini-disk of the lower mass secondary MBH, which is expected to capture a larger fraction of the mass accretion from the circumbinary disk. Figure 17 illustrates that in the system with the orbital period \(P = 1\,\)yr (or equivalently, \(a = 100\,M\)) the circumbinary and mini-disk emit a comparable amount of power, albeit at different frequencies. In the more compact MBHB system with \(P = 0.1\,\)yr (\(a = 22\,M\)), the luminosity of the mini-disk corresponds to only \(\sim 1\%\) of that of the circumbinary disk. This is contrary to simple expectations based on the efficiency of dissipation and to those laid out at the beginning of this section. More specifically, it indicates that at small enough binary separations (\(a < 100\,M\)), the reduced mass of the mini-disk leads to a diminished luminosity of its thermal emission, despite a higher efficiency of dissipation relative to the circumbinary disk.

Fig. 17
figure 17

Image(s) adapted from [left] Tanaka et al. (2012), copyright by the authors; and [right] from d’Ascoli et al. (2018), copyright by AAS

Left: Thermal luminosity spectrum of a non-spinning \(10^9\,M_\odot \) MBHB with a mass accretion rate \(3\, {\dot{M}}_E\) through the circumbinary disk. The top (bottom) panel corresponds to a binary with an orbital period \(P = 1\,\)yr (0.1 yr) or equivalently \(a = 100\,M\) (\(22\,M\)). Dotted lines show the spectrum from the circumbinary disk (peaking at lower frequencies) and the secondary mini-disk (higher frequencies). Both panels show the combined spectrum (solid thick lines) and the corresponding AGN spectrum for an Eddington-limited thin disk around a single MBH with the same mass (thin dashed). Right: Luminosity spectrum of a non-spinning \(10^6\,M_\odot \) MBHB with mass accretion rate through the circumbinary disk \(0.5\, {\dot{M}}_E\). The spectrum is calculated for a binary inspiraling from \(a=20\,M\) and is time-averaged over one orbit. It includes contributions from the mini-disk regions, accretion streams and the circumbinary disk.

Because the time-variable, quasi-periodic emission associated with accreting MBHBs is more likely to be associated with the mini-disks than the circumbinary disk, the relative dimness of the mini-disks brings into question the ability to identify it in observations. For this reason, it is important to account for all anticipated components of emission, in addition to the thermal spectrum. For example, outside of the thermalized regions the mini-disks are also expected to be significant sources of coronal emission, where inverse Compton scattering between photons and high-energy electrons gives rise to the hard X-ray emission. The right panel of Fig. 17 shows an example of the luminosity spectrum calculated from a general relativistic MHD simulation where the mini-disks, the accretion streams and the circumbinary disk are sources of both thermal and coronal emission (d’Ascoli et al. 2018). In this case, the emission is associated with a \(10^6\,M_\odot \) binary, whose mass accretion rate through the circumbinary disk is \({\dot{M}} = 0.5\, {\dot{M}}_E\), evolving from a separation of \(a=20\,M\).

The total spectrum is reminiscent of classical AGNs powered by single MBHs and exhibits two peaks: one produced by the thermal emission at the UV/soft X-ray frequencies and the other produced by coronal emission in hard X-rays. In the UV/soft X-ray band, the luminosity of the circumbinary disk is comparable to or larger than that of the two mini-disks. The hard X-ray luminosity is on the other hand dominated by emission from the mini-disks by more than an order of magnitude. The emission from the accretion streams is subdominant across the entire range of frequencies. d’Ascoli et al. (2018) find that in their simulated scenario, \(\sim 65\%\) of the emission luminosity comes from the circumbinary disk and \(\sim 25\%\) from the mini-disks. While the exact percentages may be subject to change in the future, this result nevertheless illustrates an important point: the bolometric luminosity and variability associated with the mini-disks may not dominate throughout the MBHB inspiral but their emission may be distinguished from that of the circumbinary disk in the X-ray band (at \(\ge 10^{17}\,\)Hz, corresponding to the rest frame energy of \(\ge 0.4\,\)keV).

We now return to an important characteristic feature of these spectra, visible in the left panel of Fig. 17 as an inflection or a “notch” that is caused by a deficiency of emission from the gap in the circumbinary disk. The diagnostic powers of the notch, which could potentially be used to identify MBHBs and place constraints on their geometry, were extensively discussed in the literature (see the references above, following the mention of a low density gap). This feature is visible in the modeled thermal spectra of accreting MBHBs with \(a \ge 100\,M\) and becomes weaker with a diminishing contribution of the mini-disks to the overall luminosity of the gas accretion flow in MBHBs with smaller separations. Indeed, in the right panel of Fig. 17, where the MBHB separation is \(a < 20\,M\), there is no apparent notch in the spectrum at \(\sim 2\times 10^{16}\,\)Hz. If this is a general trend in MBHBs surrounded by circumbinary flows, it suggests that the spectral notch may be a more effective diagnostic tool for more widely separated binaries, where it would be visible in the IR/optical/UV part of the spectrum. On the flip side, if either the notch or the periodically varying X-ray emission are distinct enough to be detected in the spectra of many AGNs, they can in principle be used as a smoking gun to identify hundreds of MBHBs with mass \(>10^7\,M_\odot \) in the redshift range \(0.5 \lesssim z \lesssim 1\) (Krolik et al. 2019).

4.1.3 Other modulation possibilities before the merger

In addition to the broadband X-ray emission discussed above, the two mini-disks may produce variable relativistically broadened Fe K\(\alpha \) emission lines with rest-frame energy 6.4 keV. This line commonly originates from the central region of MBH accretion flows (within \(\sim 10^3\,M\)). It is prominent in the X-ray spectrum due to the high abundance and high fluorescence yield of iron, making it easy to identify even if it is strongly distorted by relativistic Doppler shifts and gravitational redshift (Fabian et al. 1989). The Fe K\(\alpha \) line has already been used to probe the innermost regions of accretion disks and infer the spin magnitudes of single MBHs in a subset of AGNs (Reynolds 2013). In the context of the MBHBs this technique can in principle be applied to binaries with orbital separations of \(\lesssim 10^3\,M\) and used to probe the circumbinary flow and the spin magnitudes of both MBHs.

The Fe K\(\alpha \) emission properties of close MBHB systems have been investigated in a handful of theoretical models that predict the shape of the composite emission lines from circumbinary accretion flows (Yu and Lu 2001; Sesana et al. 2012; McKernan et al. 2013; Jovanović et al. 2014). Figure 18 shows a sequence of the Fe K\(\alpha \) line profiles as a function of the binary orbital phase calculated for an equal-mass binary with two mini-disks of equal luminosity. This and other models illustrate that the Fe K\(\alpha \) emission-line profiles can vary with the orbital phase of the binary and be distinct from those observed in the single MBH systems. If so, they warrant further investigation as a potentially useful MBHB diagnostic. For example, in the case of the MBHBs targeted by the PTAs, the combination of the Fe K\(\alpha \) and GW signatures could provide a unique way to learn about the properties of MBHBs, since GW alone will not place strong constraints on the binary parameters (Arzoumanian et al. 2014; Shannon et al. 2015; Lentati et al. 2015). In the case of the inspiraling MBHBs detectable by the LISA observatory (Klein et al. 2016), the constraints on the orbital and spin parameters obtained from the two messengers could provide two independent measurements that can be combined to increase the precision of the result. The prospects for identifying Fe K\(\alpha \) profile signatures of MBHBs will be greatly enhanced by future observatories such as Athena (Nandra et al. 2013) and XRISM (XRISM Science Team 2020), which will be equipped with X-ray micro-calorimeters to enable very high resolution X-ray spectroscopy.

Fig. 18
figure 18

Image adapted from Jovanović et al. (2014), copyright by COSPAR

Evolution of a composite Fe K\(\alpha \) profile from an equal-mass MBHB, showing several different orbital phases. The profiles were calculated for two slowly spinning MBHs with orbital separation \(a\approx 2000\,M\), eccentricity \(e=0.75\) and mini-disks aligned with the binary orbital plane. The horizontal axis shows the ratio of the observed and rest frame frequency.

Special and general relativistic effects could also lead to the modulation of the observed EM emission, even in MBHB systems without any intrinsic variability (like the oscillating mini-disks and sloshing streams discussed earlier). For example, Bode et al. (2010) modeled inspiraling MBHBs immersed in hot accretion flows with a smooth and continuous density distribution of the gas. They find that quasi-periodic EM signatures can still arise as a consequence of shocks produced by the MBHB combined with the effect of relativistic beaming and Doppler boosting. An object for which the signature of relativistic Doppler boosting has been modeled in some detail is a quasar PG 1302-102, whose periodic light curve led to a suggestion that it may harbor a MBHB with the rest-frame orbital period of about 4 years (Graham et al. 2015b). In this context D’Orazio et al. (2015) showed that the amplitude and the sinusoid-like shape of its light curve can be explained by relativistic Doppler boosting of emission from a compact, unequal-mass binary with separation 0.007–0.017 pc. While signatures of relativistic beaming and Doppler boosting remain of interest for inspiraling MBHBs in general (see Charisi et al. 2021), the binary candidacy of PG 1302-102 was called into question by Liu et al. (2018), who found that the evidence for periodicity decreases when new data points are added to the light curve of this object. They note that for genuine periodicity one expects that additional data would strengthen the evidence, and that the decrease in significance may therefore be an indication that the binary model is disfavored.

It is worth noting that relativistic beaming and Doppler boosting could also lead to non-axisymmetric irradiation of the accretion flow outside of the MBHB orbit. The impact of this effect is not entirely obvious: for example, it could simply involve reprocessing of the beamed radiation, or it could have structural effects, if photo-heating and radiation pressure inflate the accretion flow and produce outflows. In either case, the natural frequencies of modulation in the EM emission of the gas that is “anchored” to the orbiting MBHs (i.e., the mini-disks) would be the binary orbital frequency and its overtones.

Because relativistic beaming and Doppler boosting are strongest in MBHB systems where the binary orbital plane is close to the line of sight, the same population could also be subject to periodic self-lensing. Namely, if at least one of the MBHs is accreting, the light emitted from its accretion disk can be lensed by the other black hole. For example, D’Orazio and Di Stefano (2018) find that for \(10^6\)\(10^{10}\,M_\odot \) binaries with orbital periods \(<10\,\)yr, strong lensing events should occur in \(\sim \) 1–10% of MBHB systems that are monitored over their entire orbital period. A similar fraction (1–3%) may also show a distinct feature (a dip) in their self-lensing flares, imprinted by the black hole shadow from the lensed hole (Davelaar and Haiman 2021a, b). If so, this may provide an opportunity to extract MBH shadows that are spatially unresolved by VLBI. Using similar assumptions, Kelley et al. (2021) find that the Rubin Observatory’s LSST could detect tens to hundreds of self-lensing binaries. A light curve of one such binary candidate, nicknamed Spikey, is shown in Fig. 19 (Hu et al. 2020). In this case the authors identify a model that is a combination of Doppler modulation and a narrow, symmetric lensing spike, consistent with an eccentric MBHB with a mass \(3 \times 10^7\,M_\odot \) and rest-frame orbital period of 418 days, seen at nearly edge-on inclination.

Fig. 19
figure 19

Image(s) reproduced with permission from Hu et al. (2020), copyright by the authors

Optical light curve for Spikey, a candidate for MBHB with gravitational lensing signature, overlaid with a maximum likelihood model accounting for Doppler boosting and self-lensing. The model uncertainty is extended to show predicted flares in 2013, 2018 and 2020.

Another modulation possibility was proposed by Kocsis and Loeb (2008), who suggested that the GWs generated by the inspiraling MBHB would ripple through the disk and induce in it viscous dissipation. Under the assumption that the ripples would be dissipated efficiently, they calculated that the energy released in this way would dominate the locally generated energy at large radii. However, the high frequency of the GWs compared to the local dynamical timescales farther out in the disk means that dissipation will be extremely inefficient. Using this more realistic assumption, Li et al. (2012a) found that the energy release from dissipated ripples is insignificant in all plausible circumstances.

4.2 Signatures during or immediately after the merger

4.2.1 Reaction of matter to dynamic gravity

Most of the gravitational wave energy produced during MBH coalescence is emitted during the merger and ringdown phases. If significant matter and/or magnetic fields are present close enough to the binary, where they can interact with a dynamically changing spacetime, there are various avenues for EM emission during these stages (these topics were also discussed in a comprehensive review of relativistic aspects of accreting MBHBs in the GW-driven regime by Gold 2019). For example, Krolik (2010) showed that if the gas density in the immediate vicinity of the MBHB is high enough to make it optically thick, its characteristic luminosity is the Eddington luminosity, independent of the gas mass. It is worth noting, however, that such predictions are subject to an important assumption: that gas in the vicinity of the MBHBs can cool relatively efficiently all the way to the merger, such that it can remain sufficiently dense and optically thick. If so, the bulk of the gas will tend to reside in a rotationally-supported, optically thick but geometrically thin (or slim, \(h/r <1\)) accretion disk around the MBHB. The necessity for this assumption stems from practical reasons, as the thermodynamics of circumbinary disks (determined by the heating and cooling processes) is computationally expensive to model from first principles and is unconstrained by observations.

In reality, the balance of heating and cooling in a circumbinary flow can be significantly altered close to merger, when the gas is expected to be permeated by energetic radiation and heated by MBHB shocks. This leaves room for an additional possibility: that radiative cooling of a flow around the MBHB is inefficient. If this is the case, such a flow would resemble hot and tenuous, radiatively inefficient accretion flows (RIAFs; Ichimaru 1977; Rees et al. 1982; Narayan and Yi 1994). These tend to be less luminous than their radiatively efficient counterparts, as well as optically thin and geometrically thick (\(h/r \gtrsim 1\)). Therefore, the radiatively inefficient and radiatively efficient gas flows represent idealized scenarios that bracket a range of physical situations in which pre-coalescence MBHBs may be found. We review them both here for completeness and note when works in the literature adopt one or the other assumption. To provide continuity with the discussion of the EM precursors to merger in previous section (Sect. 4.1), here we consider the properties of MBHBs that evolve from orbital separations of about \(10\,M\) through merger and ringdown.

(a) Mergers in radiatively inefficient gas flows. In RIAFs, most of the energy generated by accretion and turbulent stresses is stored as thermal energy in the gas, and the accretion flow is hot and geometrically thick (Ichimaru 1977; Narayan and Yi 1994). The electron and ion plasmas in RIAFs can form a two-temperature flow in which the thermal energy is stored in the ion plasma while the electron plasma cools more efficiently (i.e., \(T_p > T_e\)). In such cases, the temperature of the plasma is represented by the ion temperature, while the characteristics of emitted radiation depend on the properties of electrons. The temperature ceiling reached by the ion plasma is determined by cooling processes such as thermal bremsstrahlung, synchrotron, and inverse Compton emission, as well as the electron-positron pair production and the pion decay resulting from energetic proton-proton collisions. Which process dominates the energy loss of the plasma depends sensitively on its density, temperature, and magnetic field strength, as well as the efficiency of coupling between ions and electrons. The latter process determines the rate with which energy can be transferred from hot ions to electrons, and consequently the ratio of their temperatures, \(\epsilon = T_e / T_p\). In order to illustrate the emission properties of these flows, we will make a simplifying assumption that \(\epsilon = 10^{-2}\) everywhere in the accretion flow. This is an idealization as \(T_e / T_p\) is expected to vary in both space and time and can have a range of values between \(\sim 10^{-2}\) and 0.1 depending on the dominant plasma processes (e.g., Sharma et al. 2007).

To understand the dependence of the gas luminosity on the properties of the system, we first consider the bremsstrahlung luminosity emitted from the Bondi radius of gravitational influence of a single MBH with mass M, \(R_{\mathrm{B}} = GM/c_s^2\), where \(c_s = (\gamma \, k T_p/m_p)^{1/2}\) is the speed of sound evaluated assuming the equation of state of an ideal gas and \(\gamma = 5/3\), for monoatomic gas. In geometrized units \(R_{\mathrm{B}} \approx 6.5\,M\,T^{-1}_{p,12}\) and

$$\begin{aligned} L_{\mathrm{brem}} \approx 6.7 \times 10^{44}\, \mathrm{erg\,s^{-1}} \epsilon ^{1/2}_{-2}\,T^{-1/2}_{p,12} (1 + 4.4\, \epsilon _{-2}\,T_{p,12})_{5.4}\,\tau _T^2\,M_8^4 \,. \end{aligned}$$

Here, \(\epsilon _{-2} = \epsilon /10^{-2}\), \(T_{p,12} = T_p/10^{12}\)K, \(M_8 = M/10^8\,M_\odot \), \(\tau _{\mathrm{T}} = \kappa _{\mathrm{T}}\,\rho \,R_{\mathrm{B}}\) is the optical depth for Thomson scattering within the Bondi sphere, \(\kappa _{\mathrm{T}}\) is the opacity to Thomson scattering, and \(\rho \) is the gas density. The subscript “5.4” indicates that the expression in the brackets is normalized to this value. Note that \(T_p \approx 10^{12}\)K (corresponding to \(kT_p \approx 100\,\)MeV) is the maximum temperature that the ion plasma can reach at the innermost stable circular orbit of the MBH if all of its gravitational potential energy is converted to thermal energy, so that \(GM m_p/r_{\mathrm{ISCO}} \approx kT_p\). Equation (30) implies a maximum bremsstrahlung luminosity that can be reached by an accretion flow as long as its optical depth \(\tau _{\mathrm{T}} \lesssim 1\) (because the plasma of this temperature is fully ionized, we consider Thomson scattering to be the dominant source of opacity in this regime). Flows with a larger optical depth would be subject to radiation pressure, which could alter the kinematics of the gas or unbind it from the MBH altogether, potentially erasing any characteristic variability and suppressing the luminosity.

If the hot accretion flow is threaded by a strong magnetic field, a significant fraction of its luminosity could be emitted in the form of synchrotron radiation. Assuming a field strength \(B \approx 10^4\,\mathrm{G}\,\beta _{10}^{-1/2}\,T_{p,12}\,\tau _T^{1/2}\,M_8^{-1/2}\), the synchrotron luminosity could reach

$$\begin{aligned} L_{\mathrm{syn}} \approx 4 \times 10^{46}\, \mathrm{erg\,s^{-1}} \beta _{10}^{-1}\, \tau _T^{2}\,M_8^4\,, \end{aligned}$$

where \(\beta = 8\pi \,p_{\mathrm{th}}/B^2 = 10\beta _{10}\) is the ratio of thermal to magnetic pressure in the gas, expected to reach values of \(1-10\) in the central regions of RIAFs (Cao 2011). The presence of the softer photons supplied in situ by synchrotron and bremsstrahlung emission would also give rise to inverse Compton radiation of similar magnitude

$$\begin{aligned} L_{\mathrm{IC}} \approx 2\, \epsilon _{-2}\,T_{p,12}\, \tau _T \,L_{\mathrm{soft}} \;, \end{aligned}$$

where \(L_{\mathrm{soft}}\) is the luminosity of soft radiation. This expression is evaluated for a thermal distribution of nonrelativistic electrons (see equation 7.22 in Rybicki and Lightman 1979) with temperature \(T_e \approx 10^{10}\,\)K and an emission region corresponding to the Bondi sphere associated with the MBH.

Where the high energy tail of protons reaches the threshold of \(kT_p \approx 100\,\)MeV an additional high energy process contributes to the radiative cooling: proton-proton collisions result in copious pion production, followed by pion decay to two \(\gamma \)-ray photons, \(p + p \rightarrow p + p + \pi ^0 \rightarrow p + p + 2\gamma \). Following Colpi et al. (1986), who calculated the \(\gamma \)-ray emission from the \(p-p\) collisions of a thermal distribution of protons in the vicinity of a single Kerr BH, we estimate

$$\begin{aligned} L_{pp} \approx 2 -13 \times 10^{40}\, \mathrm{erg\,s^{-1}} T_{p,12}\,\tau _T^2\,M_8 \,, \end{aligned}$$

where the two extreme values correspond to a static and maximally rotating BH, respectively. We expect the luminosity in the MBHB system to be closer to the higher value because the gas in the rotating and dynamic spacetime of the pair of orbiting black holes is very efficiently shock-heated to 100 MeV. The emission of \(\gamma \)-rays due to pion decay is strongly suppressed in the limit \(\tau _T \gtrsim 1\) due to the increased cross-section of \(\gamma \)-ray photons to electron-positron pair production, as well as the increased coupling between electron and proton plasma, which lowers \(T_p\) below the energy threshold for pion production (Colpi et al. 1986). In calculating luminosities in this section, we assumed the gas to be optically thin within the Bondi radius, which sets an upper limit on the gas density of the hot accretion flow (such that \(\rho < 2.6\times 10^{-12}\mathrm{g\,cm^{-3}} \tau _T\,M_8^{-1} \)), and thus an upper limit on \(L_{\mathrm{brem}}\), \(L_{\mathrm{syn}}\), \(L_{\mathrm{IC}}\), and \(L_{pp}\).

The spectral energy distribution of these sources would be similar to a group of low-luminosity AGNs to which RIAF models have been applied (see Nemmen et al. 2006, for example). Spectral bands where these emission mechanisms are expected to peak in the reference frame of the binary are submillimeter (synchrotron), UV/X-ray (inverse Compton), \(\sim \) 100 keV–1 MeV \(\gamma \)-ray (bremsstrahlung and inverse Compton), and \(\sim \) 20 MeV (pion decay). Additional components could in principle arise and overtake the emission from the hot gas, such as wide-band non-thermal synchrotron emission (if active and persistent jets are present in the system), as well as the optical/UV emission associated with the accretion disk that may encompass the hot flow at larger radii. Now that we laid out the expectations for emission mechanisms and luminosity of RIAFs, we turn our attention to characteristics of such flows identified in simulations of MBHB mergers.

MBHB mergers in radiatively inefficient accretion flows have been explored in simulations by multiple groups. The early work by van Meter et al. (2010b) examined test particle orbits in the presence of a coalescing binary and concluded that there could be collisions at moderate Lorentz factor, which might lead to high-energy emission. A few subsequent works explored the fluid and emission properties of binary accretion flows resembling RIAFs in the context of general relativistic hydrodynamic simulations (Bode et al. 2010, 2012; Farris et al. 2010; Bogdanović et al. 2011). They found that because of high thermal velocities, the radial inflow speeds of gas in the flow are comparable to the orbital speed at a given radius. This implies that in a hot gas flow, unlike the circumbinary disk scenario, binary torques are incapable of creating a central low density region, because the gas ejected by the binary is replenished on a dynamical timescale. As a consequence, the MBHB remains immersed in the flow until merger, which allows it to interact continually with the fluid and shock it to temperatures close to \(\sim 10^{12}\,\)K. Figure 20 shows that the shocks are initially confined to the tidal wakes following the two MBHs. Around the time of the merger, the shocked, high temperature gas is promptly accreted by the newly formed daughter MBH.

Fig. 20
figure 20

Image adapted from Farris et al. (2010), copyright by APS

Contours tracing the amplitude of shock-heating of the gas in vicinity of the MBHB at \(t = 14\,\mathrm{days}\, M_8\), \(3\,\mathrm{days}\,M_8\), \(2\,\mathrm{h} \,M_8\) before and \(2\,\mathrm{days}\,M_8\) after the merger, respectively. Shocks are initially confined to the tidal wakes following the two MBHs on their orbit. The shocked gas is promptly accreted by the newly formed daughter MBH. Arrows denote velocity vectors.

Both the appearance and disappearance of shocks give rise to a characteristic variability of the emitted light. This is illustrated in the left panel of Fig. 21, which shows bremsstrahlung luminosity as a function of time, normalized by the light curve for a single MBH with equivalent mass (see Eq. (30)). The most characteristic feature is the broad peak in luminosity, whose growth coincides with the formation of the shocked region within the binary orbit. As the binary shrinks, the brightness of this region increases until the merger, at which point the final MBH swallows the shocked gas and the luminosity drops off precipitously.

Fig. 21
figure 21

Image adapted from Bode et al. (2012), copyright by APS

Left: Bremsstrahlung luminosity of hot accretion flows for different MBHB configurations, including mass ratios \(q=1\) (CH1) and 1/2 (CH2, CH3, CH4) and different spin orientations (CH1, CH2 – parallel to the orbital angular momentum; CH3, CH4 – misaligned), normalized to the luminosity of a single MBH with corresponding mass. Coalescence occurs at \(t=0\). Right: Closer inspection of the MBHB light curves on the left reveals a correlation between the EM signal (red, dotted line) and gravitational waveforms (blue, solid) during the inspiral, shown for a particular model (CH2) but present in all models to some degree. The quasi-periodic signal in the EM emission is obtained as a ratio of the beamed to unbeamed light curve and is calculated for a fiducial observer placed in the plane of the binary at infinity.

Interestingly, Bode et al. (2010, 2012) showed that in such systems, the changing beaming pattern of the orbiting binary surrounded by emitting gas can also give rise to modulations in the observed luminosity of the EM signal that is closely correlated with GWs (also referred to as the “EM chirp”). This can be seen in the right panel of Fig. 21, which shows the quasi-periodic signal in the EM emission (obtained as a ratio of the beamed to unbeamed light curve) superposed on the GW emission in arbitrary units. It is worth noting that the amplitude of the EM variability depends sensitively on the thermodynamic properties of the gas disk and optical depth of the surrounding gas. In order for it to be identified as a unique signature of an inspiraling and merging MBHB in observations, the EM chirp would need to have a sufficiently large amplitude, probably comparable to or larger than the intrinsic variability of non-binary AGNs (which for example corresponds to \(\sim 10\%\) in X-rays).

The signatures discussed above are somewhat weaker for lower mass ratios, because orbital torques from an unequal-mass binary are less efficient at driving shocks in the gas, resulting in a less luminous, shorter lasting emission from this region. Similarly, in systems with generic spin orientations (CH3 and CH4), the luminosity peaks at lower values, relative to the \(q = 1\) aligned binary. This is a consequence of the orbital precession present in binaries with misaligned spins, which further inhibits the formation of a stable shock region between the holes. The gradual rise and sudden drop-off in luminosity, however, seem to be a generic feature of all modeled light curves, regardless of the spin configuration and the mass ratios. They were observed for a relatively wide range of initial conditions (Bode et al. 2010; Farris et al. 2010), indicating that this is a robust signature of binary systems merging in hot accretion flows.

(b) Mergers in radiatively efficient gas flows. As noted in Sect. 4.1.1, simulations of inspiraling MBHBs indicate that the inner rim of the circumbinary disk is able to follow the binary to radii of \(\sim 10\,M\) (Noble et al. 2012; Farris et al. 2015a; Bowen et al. 2017; Tang et al. 2018). If matter can keep up with the binary to even smaller separations, then the EM signatures would be a smooth extension of those discussed in the previous section (associated with the time-variable disks and streams). By extension, the spectra associated with gas that emits until the instant of merger would also be qualitatively similar to those shown in Fig. 17, namely, a combination of thermal and coronal emission.

Figure 22 shows the EM light curves and the GW signal calculated from one such system, where the initial separation of the binary is \(a=10\,M\). The emission signatures shown in the figure were calculated from the first fully relativistic 3D hydrodynamic simulation of an equal-mass, non-spinning MBHB coalescing in a geometrically thin circumbinary disk with the initial scale height \(h/r \sim 0.1\) (Farris et al. 2011). The most notable properties of the displayed EM light curves is that their luminosity decreases as a consequence of late-time decoupling of the MBHB from the circumbinary disk and that their variability does not trivially correspond to the GW signal.

Fig. 22
figure 22

Image adapted from Farris et al. (2011), copyright by APS

Left: Synchrotron (solid line) and bremsstrahlung (dashed, relatively flat line) luminosity associated with an equal-mass, non-spinning MBHB coalescing in a circumbinary disk. The synchrotron luminosity was calculated for \(\beta =10\). The luminosity unit is \({\hat{L}} = L/[10^{46}\,M_8^3\,n_{12}^2\,\mathrm{erg\,s^{-1}}]\), where \(n_{12} = n/10^{12} \mathrm{cm^{-3}}\) is the gas number density. Right: \(h_+\) is the \(+\) polarization of the GW signal measured by an observer looking down the polar axis at a distance D from the MBHB. The initial separation of the binary is \(a=10\,M\) and coalescence occurs at \(t=0\).

A more optimistic conclusion was reached by Tang et al. (2018), who used 2D viscous hydrodynamic simulations of an equal-mass MBHB surrounded by a radiatively efficient circumbinary disk with the assumed scale height \(h/r = 0.1\). By evolving the inspiral of the MBHB from \(a=60\,M\) to merger, they find that it continues to accrete efficiently, and thus remains luminous all the way to merger. Furthermore, Tang et al. (2018) find that these systems display strong periodicity at twice the binary orbital frequency throughout the entire inspiral. The quasi-periodic emission, modeled as modified blackbody radiation, is most pronounced in the X-ray band and associated with strong shocks at the inner rim of the circumbinary disk (at \(\sim 2\,\)keV) and the two mini-disks (\(\sim 10\,\)keV). Because a clear EM chirp, correlated with the GW emission, is present until the very end of inspiral and the EM emission can potentially reach Eddington-level luminositiesFootnote 13, these types of systems would represent the most promising multimessenger MBHBs.

In addition to the results themselves, one can also appreciate the range of predictions about the EM signatures found in simulations described above and more generally, in the literature, that stems from differences in the simulation setup: initial conditions, dimensionality and other technical details. They hint at a complex nature of these models, which are still work in progress, as well as the need to carefully examine the dependence of the EM signatures on additional physical phenomena, such as the magnetic fields and radiation.

4.2.2 Effect of dynamical spacetime on magnetic fields and formation of jets

We now discuss the response of magnetic fields to the dynamical spacetime of a merging MBHB. In order to separate the effects of the spacetime and the gas on magnetic fields, it is useful to examine mergers in three different scenarios: (a) in (near) electrovacuum, (b) in radiatively inefficient, magnetized gas flows resembling RIAFs, and (c) in radiatively efficient, magnetized circumbinary disks.

(a) Mergers in (near) electrovacuum. The earliest group of papers on this topic explored the effect of the dynamical spacetime on pure magnetic fields in a vacuum (Palenzuela et al. 2009; Mösta et al. 2010) and on magnetic fields within a tenuous plasma with zero inertia (in a so-called force-free approximation Palenzuela et al. 2010a, b, c; Mösta et al. 2012; Alic et al. 2012). These setups are illustrative of conditions that might arise if the binary decouples from the circumbinary disk early in the inspiral but continues to interact with the fields anchored to it. They were the first to indicate that even if the gas is dilute and optically thin, the winding of the magnetic fields by the MBHB could lead to Poynting outflows and the formation of jets.

This is illustrated in Fig. 23, which shows the Poynting flux associated with the electric and magnetic fields just hours before and after the coalescence of two non-spinning MBHs (Palenzuela et al. 2010b). This and other studies established that the winding of magnetic fields by the binary results in the enhancement of the magnetic energy density and the Poynting luminosity on the account of MBHB kinetic energy. The Poynting luminosity scales as

$$\begin{aligned} L_{\mathrm{Poynt}}\propto & {} \,v^2\, B_0^2\, M^2 \,\,\, \mathrm{and} \end{aligned}$$
$$\begin{aligned} L_{\mathrm{Poynt, peak}}\approx & {} \,3\times 10^{43}\,\mathrm{erg\,s^{-1}} \left( \frac{B_0}{10^4\,\mathrm{G}}\right) ^2 \left( \frac{M}{10^8\,M_\odot }\right) ^2 \,\,, \end{aligned}$$

where v is the orbital speed of the MBHB, \(B_0\) is the strength of the initial, uniform magnetic field and \(L_{\mathrm{Poynt, peak}}\) is the peak value of the Poynting luminosity. Given that v reaches maximum around merger, these works find that the Poynting luminosity also peaks at merger, as shown in the left panel of Fig. 24. They also make other important points: (i) even if all of the electromagnetic energy is spent on charge acceleration and eventually reradiated as photon energy, the emitted luminosity is expected to be orders of magnitude lower than Eddington (assuming \(B_0\sim 10^4\,\mathrm{G}\); Mösta et al. 2010) and (ii) the Poynting luminosity associated with the collimated dual-jet structure is \(\sim 100\) times smaller than that emitted isotropically, making the detection of jets less likely (Mösta et al. 2012; Alic et al. 2012). This led to some early reservations about a detectability of the merger flare associated with the dual jet.

Fig. 23
figure 23

Image adapted from Palenzuela et al. (2010b), copyright by AAAS

Poynting flux associated with the electromagnetic field just hours before and after the MBH coalescence. From left to right: \(t=-11\mathrm{h}\, (M/10^8\,M_\odot )\), \(t=-3.0\mathrm{h}\, (M/10^8\,M_\odot )\), \(t=4.6\mathrm{h}\, (M/10^8\,M_\odot )\), and \(t=6.8\mathrm{h}\, (M/10^8\,M_\odot )\), all expressed in the frame of the MBHB. Darker colors mark larger flux amplitudes.

(b) Mergers in radiatively inefficient, magnetized gas flows. More encouraging results were found by general relativistic magnetohydrodynamic (GRMHD) simulations of MBHBs immersed in uniform density plasma threaded by an initially uniform magnetic field (Giacomazzo et al. 2012; Kelly et al. 2017; Cattorini et al. 2021). Equivalently to the relativistic hydrodynamic simulations with uniform gas flows discussed in the previous section, these setups correspond to physical scenarios in which the magnetized gas flows inwards as a RIAF. The key finding of these simulations is that in the presence of gas, increase in the magnetic field strength and energy density is even more dramatic relative to the simulations of magnetic fields in (near) vacuum. This results in a commensurate increase in the Poynting luminosity, which according to Kelly et al. (2017) scales as

$$\begin{aligned} L_{\mathrm{Poynt}}\propto & {} \,\rho _0\, v^{2.7} M^2 \,\,\, \mathrm{and} \end{aligned}$$
$$\begin{aligned} L_{\mathrm{Poynt, peak}}\approx & {} \,1.2\times 10^{46}\,\mathrm{erg\,s^{-1}} \left( \frac{\rho _0}{10^{-13}\,\mathrm{g\,cm^{-3}}}\right) \left( \frac{M}{10^8\,M_\odot }\right) ^2 \,\,, \end{aligned}$$

where \(\rho _0\) corresponds to the initial, uniform value of gas density, v is the orbital speed of the MBHB, and \(L_{\mathrm{Poynt, peak}}\) is again the peak value of the Poynting luminosity. A comparison with Eq. (35) reveals several important differences. One is that in the presence of gas, the resulting Poynting luminosity does not depend on the initial strength of the magnetic field (this outcome was tested for field strengths in the range \(B_0 = 3\times 10^3 - 3\times 10^4\,\mathrm{G}\); Kelly et al. 2017). The other is that the magnetic field strength exhibits super-quadratic growth on approach to merger, due to the accretion of gas, which further compresses the field lines near the horizon. As a result, the peak value of the Poynting luminosity is orders of magnitude higher than that inferred from simulations in near vacuum.

The right hand side of Fig. 24 shows the evolution of the Poynting luminosity found in this GRMHD simulation. After the initial magnetic field configuration settles into a quasi-steady state during the MBHB inspiral, \(L_{\mathrm{Poynt}}\) gradually rises and exhibits a local peak around the time of the merger (marked as “(d) merger blip" in the figure). This local peak is equivalent to the peak in luminosity evident in the left hand panel, found in simulations of MBHBs and fields in near vacuum. The most striking difference between the two scenarios is that in near vacuum the Poynting luminosity rapidly decays to the level consistent with that associated with a single spinning MBH, whereas in magnetized plasma it reaches a new post-merger maximum. The post-merger increase in \(L_{\mathrm{Poynt}}\) to the level of the Eddington luminosity (\(L_E = 1.3\times 10^{46}\,\mathrm{erg\,s^{-1}} M_8\)) is a result of continued accretion onto the remnant, spinning MBH which leads to a late increase in magnetic field strength.

Fig. 24
figure 24

Image(s) adapted from [left] Palenzuela et al. (2010b), copyright by AAAS; and [right] from Kelly et al. (2017), copyright by APS

Left: Poynting luminosity as a function of time (red, solid line) associated with the MBH coalescence in near vacuum, calculated for \(M=10^8\,M_\odot \) MBHB and \(B_0=10^4\,\mathrm{G}\). The blue dashed line and axis on the right mark the corresponding GW luminosity. Time \(t=0\) corresponds to merger. Right: Poynting luminosity associated with the MBH coalescence in a radiatively inefficient magnetized gas flow. The vertical dashed line corresponds to the instant of merger. For \(M=10^8\,M_\odot \) and \(\rho _0 = 10^{-13}\,\mathrm{g\,cm^{-3}}\), the units of time and luminosity correspond to \(1 \, t = 0.14\,\mathrm{h}\,\, M_8\) and \(1 \, L_{\mathrm{Poynt}} = 5.9\times 10^{44}\,\mathrm{erg\,s^{-1}}\,\rho _{-13}\,M_8^2\).

All studies of MBHB mergers in radiatively inefficient, magnetized gas flows discussed here investigated equal mass binaries. This is a configuration which maximizes the winding and compression of magnetic field lines, and we thus expect it to result in the highest Poynting luminosities (see also the discussion of mergers in magnetized circumbinary disks next). The exact dependence of the Poynting luminosity on the MBHB mass ratio is yet to be determined in future studies. The dependance of the Poynting luminosity on the MBH spins was recently studied by Cattorini et al. (2021), who examined configurations with equal spins and dimensionless magnitudes \(s_1 = s_2 = 0.3\) and 0.6, parallel to the orbital angular momentum. They find that the peak luminosity, reached shortly after the merger (defined as “(d) merger blip” in Fig. 24), depends on the MBH spins and is enhanced by a factor of about 2 (about 2.5) for binaries with spin magnitudes 0.3 (0.6) relative to the nonspinning binaries.

Using a scaling with physical properties similar to Eq. (31) and the assumption that the gas is optically thin to its own emission, Kelly et al. (2017) also estimated the synchrotron luminosity associated with a MBHB immersed in magnetized, initially uniform-density plasma. They found that the shape of the calculated light curve is qualitatively similar to that shown in the left panel Fig. 21, i.e., the synchrotron luminosity gradually increases until the merger, at which point it drops off precipitously. Figure 25 illustrates the broad-band spectrum associated with this type of accretion flow, corresponding to the instant of super-Eddington luminosity at the peak of the EM and GW emission. The spectrum was calculated by Schnittman (2013), who applied relativistic Monte-Carlo ray-tracing to the GRMHD simulation by Giacomazzo et al. (2012) as a post-processing step. In this approach, the “seed” synchrotron and bremsstrahlung photons produced by the plasma are ray-traced through gravitational potential of the MBHB. They are allowed to scatter off the hot electrons in a diffuse corona (similar to the AGN coronae), giving rise to a power-law spectrum with cut-off around \(kT_e = 100\,\mathrm{keV}\), characteristic of inverse Compton scattering.

Fig. 25
figure 25

Image(s) reproduced with permission from Schnittman (2013), copyright by IOP

Broad-band spectrum inferred from the GRMHD simulation of a MBHB immersed in magnetized, uniform density plasma by Giacomazzo et al. (2012). The spectrum is evaluated at the instant in time near the peak of the EM and GW emission and it corresponds to \(10^7\,M_\odot \) MBHB, the gas with \(kT_e = 100\,\mathrm{keV}\) and optical depth \(\tau _T \approx 1\).

(c) Mergers in radiatively efficient, magnetized circumbinary disks. It is interesting to compare these results to the mergers of MBHBs immersed in magnetized disks. This setup was investigated by Farris et al. (2012), Gold et al. (2014a, b), Paschalidis et al. (2021) and (Combi et al. 2021). Again in this case, the studies show that the winding of magnetic field lines leads to the formation of a collimated dual jet and an increase in the magnitude of the Poynting luminosity after merger (see Fig. 26). The main difference from mergers in RIAFs is that the MBHB decouples from the circumbinary disk in the late stages of inspiral and just before merger (when \(a < 10\,M\) in geometrically thick disks with \(h/r \approx 0.3\)). This leads to a decrease in the accretion rate and luminosity of the gas flow just before merger, which is most pronounced for equal-mass binaries (Gold et al. 2014b). This is qualitatively consistent with the behavior observed in early simulations of unmagnetized disks (Farris et al. 2011; Bode et al. 2012) and discussed in Sect. 4.2.1. The diminished gas flow provides weaker compression and collimation of magnetic field lines and leads to a less dramatic increase in the Poynting luminosity than in the case of RIAFs.

Fig. 26
figure 26

Image(s) reproduced with permission from Gold et al. (2014b), copyright by APS

Instants before (left panels) and after (right panels) merger of a MBHB surrounded by a geometrically thick, magnetized circumbinary disk. The winding of magnetic field lines leads to the formation of a collimated jet. However, the diminished gas flow close to merger (visible in the bottom panels) provides weaker compression of magnetic field lines and leads to a more modest increase in the Poynting luminosity than in the case of RIAFs. The scale of the top panels is about \(150\,M\). The bottom panels provide a closeup view within \(20\,M\).

The dependence of the \(L_{\mathrm{Poynt, peak}}\) on the MBHB mass ratio was explored in Gold et al. (2014b), who found that it decreases with decreasing q, because the equal-mass setup maximizes the pre-merger winding of the magnetic field lines as well as the post merger gas inflow. Furthermore, the results of studies that consider mergers of spinning MBHs, with dimensionless spin magnitudes \(s_1 = s_2 = 0.6\), parallel to the orbital angular momentum, show only modest enhancement (less than 50%) in the \(L_{\mathrm{Poynt, peak}}\) as a consequence of additional winding of magnetic field lines by the individual MBH spins before merger (Mösta et al. 2012; Alic et al. 2012). This is consistent with more recent findings by Paschalidis et al. (2021) and Combi et al. (2021), based on the GRMHD simulations of late inspiral in binaries with \(a\lesssim 20\,M\). They report the enhancement of the Poynting flux in MBHB systems with \(s_1 = s_2 = 0.75\) and 0.6, respectively, aligned with the orbital angular momentum. In addition to these general trends, Combi et al. (2021) find that regardless of the spin magnitudes, late in the inspiral the Poynting flux is modulated at the beat frequency between the MBHB orbital frequency and the frequency of an overdense lump, driven by the quasi-periodicity of accretion onto the holes themselves. This is interesting because it indicates that quasi-periodicity may also be present in the emission from the jets associated with inspiraling MBHBs, if a sufficient amount of the Poynting flux is tapped to power acceleration of charged particles in the jets and their subsequent EM emission.

All studies discussed in this section (Sect. 4.2.2) examine configurations in which the pre-merger MBH spins are zero, aligned or anti-aligned, and in which the fields lines are initially either poloidal or parallel to the orbital angular momentum of the binary, and thus to the spin axis of the remnant MBH. This is the setup that maximizes the pre-merger winding and enhancement of magnetic fields, and it is plausible that more tangled and intermittent initial fields may lead to lower \(L_{\mathrm{Poynt, peak}}\) than the reported values. For example, Palenzuela et al. (2010a) and Kelly et al. (2021) find that the steady-state (peak) Poynting luminosity depends strongly on the initial field angle with respect to the remnant MBH spin axis, with maximum luminosity achieved for perfect alignment and minimum luminosity achieved for the field perpendicular to the remnant’s spin. Furthermore, they report that the proto-jet is formed along the remnant MBH spin-axis near the hole, while aligning with the asymptotic magnetic field at large distances. This raises interesting questions about what astrophysical processes set the geometry of magnetic fields on larger scales, beyond the influence of the MBH spin, and what is the resulting jet geometry in systems in which the remnant MBH undergoes a spin “flip" relative to the spin (and possibly jet) axes of the pre-merger MBHs.

In summary, the GRMHD studies of MBHB mergers in magnetized environments carried out so far indicate that during and immediately after the merger \(L_{\mathrm{gas}} < L_{\mathrm{Poynt}}\) in the radiatively inefficient flows, whereas \(L_{\mathrm{Poynt}} < L_{\mathrm{gas}}\) in the radiatively efficient flows. The peak total luminosity (\(L_{\mathrm{gas}} + L_{\mathrm{Poynt}}\)) in either physical scenario can be comparable to, or larger than the Eddington luminosity. Thus, an opportunity for the EM luminous emission from jets exists, given that a sufficient fraction of the Poynting power is converted into charged particle acceleration and subsequently, photon luminosity. The picture emerging from these works is that radiatively inefficient accretion flows provide the best opportunity for collimation and enhancement of magnetic fields, and consequently for formation of the EM luminous jets coincident with the merger. In radiatively efficient accretion flows the luminous EM jets may still form but with some delay, corresponding to the viscous timescale for circumbinary gap refilling. In this case, the nonthermal EM emission from electrons accelerated by the forward shock in jets may emerge days to months after the MBH coalescence and appear as a slowly fading transient (Yuan et al. 2021). According to the same study, the multiwavelength emission from systems with post-merger accretion rates at the Eddington level (and thus, the EM luminous jets) may persist for months and be detectable out to \(z\sim 5-6\) in the radio band and to \(z\sim 1-2\) in the optical and X-ray bands.

Besides the EM emission, the relativistic jets launched close to the MBH coalescence can also produce neutrino counterpart emission originating from cosmic rays accelerated in the jet-induced shocks. For example, Yuan et al. (2020) find, under somewhat optimistic assumptions, that the post-merger high-energy neutrino emission (\(E_\nu \gtrsim 1\,\)PeV) from individual GW events may be detectable by the next generation IceCube-Gen2 detector, within 5-10 years of observation. If so, a truly multiple messenger detection of MBH coalescences may be possible in some cases.

4.2.3 Retreat of the disk due to mass-energy loss

When two black holes of comparable mass merge, several percent of the mass-energy of the binary is radiated in GWs (e.g., Boyle and Kesden 2008; Tichy and Marronetti 2008; Kesden 2008; Reisswig et al. 2009; Lousto et al. 2010; Barausse et al. 2012). Most of the radiation is emitted in the last orbit or so, which is much shorter than the orbital timescale of the inner edge of the circumbinary disk. Thus, from the standpoint of the disk, the mass-energy loss occurs impulsively, causing the disk orbits to adjust to the new central potential (Bode and Phinney 2007).

As discussed by O’Neill et al. (2009), the most immediate consequence of the mass-energy loss is that the inner edge of the circumbinary disk moves outward, because the matter in the disk maintains its angular momentum but suddenly finds itself around a lower-mass object. Thus, its original radial location now becomes the pericenter of an eccentric orbit. The gas therefore oscillates in radius, with different radii having different oscillation periods. Bode and Phinney (2007) suggested that the radial gradient in radial epicyclic period would lead to shocks between annuli that would enhance the emission in some energy bands. The oscillation of gas in radius is illustrated in Fig. 27, from O’Neill et al. (2009), who explored this scenario with 3D hydrodynamic and MHD simulations. They found that the most observable effect was likely to be a decrease in the accretion rate and luminosity of the system, followed by a gradual recovery to the original level of accretion and emission on a characteristic (viscous) timescale, as seen in Fig. 28.

Fig. 27
figure 27

Image adapted from O’Neill et al. (2009), copyright by AAS

Snapshots of gas density showing oscillation of the accretion disk around the remnant MBH after coalescence, caused by the mass-energy loss of 10%, assumed to be associated with the GW emission. The mass-loss event takes place at \(t=0\). The initial wave is visible at \(t=134\,M\), while the stronger shock is visible at \(t=384\,M\) and \(1384\,M\). Shocks and rarefactions propagate radially outward in the disk (from left to right in these images). The size of each panel corresponds to about \(50\,M\) in height and radius. The colorbar is logarithmic, ranging from \(10^{-4}\) (dark) to 1 (light) in units of the initial gas density.

To explore this suggestion, we follow O’Neill et al. (2009) in noting that the magnitude of the reaction of the circumbinary disk depends strongly on how relativistic the disk is. Starting with a Newtonian disk, we suppose that the fractional mass-energy loss during merger and ringdown is \(\epsilon \ll 1\), so that if the original mass-energy was \(M_0\), the mass-energy after ringdown is \(M=M_0(1-\epsilon )\). A fluid element in the disk at radius r has a specific angular momentum of \(\ell _0=\sqrt{GM_0r}\) both before and after the merger, such that if it circularizes at constant angular momentum, its new radius and semimajor axis will be \(a=r(1+\epsilon )\) to first order in \(\epsilon \). Because its pericenter distance is \(r=a(1-e)\), the initial eccentricity of the orbit is \(e=\epsilon \) to first order. The energy release can be computed from the general Newtonian formula for the angular momentum of an orbit with eccentricity e: \(\ell (e)=\sqrt{GMa(1-e^2)}\). This implies that during circularization the fractional energy release will be \(\varDelta E/E_{\mathrm{bind}}=e^2=\epsilon ^2\), where \(E_{\mathrm{bind}}\) is the binding energy at r. Given the value of \(\epsilon \sim 0.05\) or less for most comparable-mass mergers, this suggests a local release of energy that is \(\sim \, {\mathrm{few}}\ \times 10^{-3}\) times the local binding energy.

Fig. 28
figure 28

Image adapted from O’Neill et al. (2009), copyright by AAS

Left: Mass accretion rate across the innermost stable circular orbit of the remnant MBH corresponding to the mass-loss values of 10% (solid line) and 1% (dot-dashed). The mass-loss event takes place at \(t=0\). The 10% mass-loss event shows an initial drop in the accretion rate, followed by a gradual resumption of accretion on the viscous timescale. Right: Bremsstrahlung luminosity measured between disk radii 5–35\(\,M\) corresponding to the same mass-loss values as in the left panel. In both cases there is barely any increase in luminosity. There is, however, a drop in luminosity followed by a gradual recovery. The initial accretion rate and luminosity prior to binary merger are shown at \(t < 0\) for reference. Both are shown in dimensionless units.

The time over which the annuli can release their energy is no shorter than the time needed for neighboring annuli to go sufficently out of phase with each other, so that their relative radial speed exceeds the sound speed (if the relative speed is subsonic, then conversion of the motion into heat is much less efficient). Given that the sound speed is h/r times the orbital speed in a Shakura–Sunyaev disk (Shakura and Sunyaev 1973), this suggests that only fairly geometrically thin disks will have low enough sound speeds for shocks to form. More specifically, the total radial distance moved during a radial epicycle is \(2\epsilon r\) in half a period, versus \(\pi r\) moved orbitally in half a period. The sound speed is \((h/r)v_{\mathrm{Kep}}\) (where \(v_{\mathrm{Kep}} \) is the circular orbital speed), so we compare \((h/r)v_{\mathrm{Kep}}\) to \((2/\pi )\epsilon v_{\mathrm{Kep}}\) to conclude that \(h/r<2\epsilon /\pi \) is a necessary criterion for formation of shocks. This implies \(h/r<1/(10\pi )\) if \(\epsilon =0.05\), suggesting a disk with an accretion rate well below the Eddington rate (Shakura and Sunyaev 1973).

The time for neighboring annuli to go out of phase by a radian is \(T \sim 0.1P_{\mathrm{orb}}(r)/\epsilon \), so the maximum specific luminosity is

$$\begin{aligned} L_{\mathrm{max}}=\frac{\varDelta E}{T}\approx 10\,\epsilon ^3 \frac{E_{\mathrm{bind}}}{P_{\mathrm{orb}}}\; . \end{aligned}$$

For example, \(\epsilon =0.05\) implies \(L_{\mathrm{max}}\approx 10^{-3} E_{\mathrm{bind}}/P_{\mathrm{orb}}\). The natural disk luminosity is

$$\begin{aligned} L_{\mathrm{disk}}\sim \frac{E_{\mathrm{bind}}}{P_{\mathrm{orb}}(r)/[3\pi \alpha (h/r)^2]} \sim 10^{-3}\, \frac{E_{\mathrm{bind}}}{P_{\mathrm{orb}}}\; , \end{aligned}$$

where the expression in the denominator corresponds to the viscous timescale at radius r and we have used \(\alpha =0.1\) for the Shakura-Sunyaev viscosity parameter and \(h/r=1/(10\pi )\) from the shock requirement above. Thus for sufficiently thin disks the modulation factor can be of order unity, but at the cost of a generally low luminosity. Note as well that if the disk is Newtonian, the binding energy at the inner edge of the disk is low (this would be the case if the binary decouples from the disk when the binary separation is of order