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:
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What is the semimajor axis at which MBHB evolution driven by stellar and gas interactions transitions to binary evolution driven by gravitational radiation?
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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.
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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.
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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.
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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.
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EM counterparts of merging MBHBs can originate from
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• The circumbinary disk.
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• Mini-disks and accretion streams.
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• Magnetized jets.
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• If all else fails: emission from the host galaxy.
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Electromagnetic radiation before the merger
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.
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
$$\begin{aligned} \varSigma < 2\pi \left( \frac{h}{r}\,\kappa _{\mathrm{T}} \right) ^{-1} \left( \frac{r_{\mathrm{in}}c^2}{GM}\right) ^{1/2}\; . \end{aligned}$$
(18)
This corresponds to a quantitative value of the surface density
$$\begin{aligned} \varSigma < 2\times 10^3~\mathrm{g~cm}^{-2} \left( \frac{h/r}{0.1} \right) ^{-1} \left( \frac{\kappa _{\mathrm{T}}}{0.4~\mathrm{cm}^2~\mathrm{g}^{-1}} \right) ^{-1} \left( \frac{r_{\mathrm{in}}}{200\,M}\right) ^{1/2} \,, \end{aligned}$$
(19)
and total mass
$$\begin{aligned} m < f\pi r_{\mathrm{in}}^2\,\varSigma \approx 30\,M_\odot \,f \left( \frac{h/r}{0.1} \right) ^{-1} \left( \frac{\kappa _{\mathrm{T}}}{0.4~\mathrm{cm}^2~\mathrm{g}^{-1}} \right) ^{-1} \left( \frac{r_{\mathrm{in}}}{200\,M}\right) ^{5/2} \left( \frac{M}{10^8~M_\odot }\right) ^2 \,. \end{aligned}$$
(20)
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)
$$\begin{aligned} L_{\mathrm{GW}}=-{32\over 5}{G^4\eta ^4M^5\over {c^5a^5(1-e^2)^{7/2}}} \left( 1+{73\over {24}}e^2+{37\over {96}}e^4\right) \; . \end{aligned}$$
(21)
Setting \(e_{\mathrm{GW}}=0.6\) in this stage of evolution, we rewrite this as
$$\begin{aligned} L_{\mathrm{GW}}=65\left( \frac{\eta }{0.25} \right) ^4 \left( \frac{GM}{ac^2}\right) ^5 \left( \frac{c^5}{G} \right) \; . \end{aligned}$$
(22)
Equating this to \((L_{\mathrm{t}}/L_{\mathrm{E}}) L_{\mathrm{E}}\) gives the transition semimajor axis
$$\begin{aligned} a_{\mathrm{GW}}\approx 370\,M \left( \frac{M}{10^8\,M_\odot } \right) ^{-1/5} \left( \frac{L_{\mathrm{t}}}{L_{\mathrm{E}}} \right) ^{-1/5}\; , \end{aligned}$$
(23)
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
$$\begin{aligned} a_{{\mathrm{yr}}}\approx 460\,M \left( \frac{M}{10^8\,M_\odot } \right) ^{-2/3}\;. \end{aligned}$$
(24)
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
$$\begin{aligned} e_{\mathrm{yr}}\approx e_{\mathrm{GW}} \left( \frac{a_{\mathrm{yr}}}{a_{\mathrm{GW}}} \right) ^{19/12}\approx 0.6 \left( \frac{M}{10^8~M_\odot } \right) ^{-133/180} \left( \frac{L_{\mathrm{t}}}{L_E} \right) ^{19/60}\; , \end{aligned}$$
(25)
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
$$\begin{aligned} t_{\mathrm{GW}} \approx 2\times 10^4\,{\mathrm{yr}}\, \left( \frac{M}{10^8~M_\odot } \right) ^{-5/3} \,, \end{aligned}$$
(26)
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
$$\begin{aligned} d_{\mathrm{L}}\approx 10\,\mathrm{Gpc}\, \left( \frac{L}{L_E}\right) ^{1/2} \left( \frac{M}{10^8~M_\odot } \right) ^{1/2} \left( \frac{F}{10^{-12}\,\mathrm{erg\,cm}^{-2}\mathrm{s}^{-1}}\right) ^{-1/2}\; . \end{aligned}$$
(27)
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).
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
$$\begin{aligned} \frac{r_2}{a(1-e)}={0.49q^{2/3}\over {0.6q^{2/3}+\ln (1+q^{1/3})}}\; . \end{aligned}$$
(28)
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
$$\begin{aligned} \frac{t_{\mathrm{visc}}}{P_{\mathrm{orb}}}\approx \left( \frac{\alpha }{0.1} \right) ^{-1} \left( \frac{h/r}{0.3} \right) ^{-2}\; . \end{aligned}$$
(29)
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.
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).
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.
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.
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.
Signatures during or immediately after the merger
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
$$\begin{aligned} L_{\mathrm{brem}} \approx 6.7 \times 10^{44}\, \mathrm{erg\,s^{-1}} \epsilon ^{1/2}_{-2}\,T^{-1/2}_{p,12} (1 + 4.4\, \epsilon _{-2}\,T_{p,12})_{5.4}\,\tau _T^2\,M_8^4 \,. \end{aligned}$$
(30)
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
$$\begin{aligned} L_{\mathrm{syn}} \approx 4 \times 10^{46}\, \mathrm{erg\,s^{-1}} \beta _{10}^{-1}\, \tau _T^{2}\,M_8^4\,, \end{aligned}$$
(31)
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
$$\begin{aligned} L_{\mathrm{IC}} \approx 2\, \epsilon _{-2}\,T_{p,12}\, \tau _T \,L_{\mathrm{soft}} \;, \end{aligned}$$
(32)
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
$$\begin{aligned} L_{pp} \approx 2 -13 \times 10^{40}\, \mathrm{erg\,s^{-1}} T_{p,12}\,\tau _T^2\,M_8 \,, \end{aligned}$$
(33)
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.
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.
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.
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.
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
$$\begin{aligned} L_{\mathrm{Poynt}}\propto & {} \,v^2\, B_0^2\, M^2 \,\,\, \mathrm{and} \end{aligned}$$
(34)
$$\begin{aligned} L_{\mathrm{Poynt, peak}}\approx & {} \,3\times 10^{43}\,\mathrm{erg\,s^{-1}} \left( \frac{B_0}{10^4\,\mathrm{G}}\right) ^2 \left( \frac{M}{10^8\,M_\odot }\right) ^2 \,\,, \end{aligned}$$
(35)
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.
(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
$$\begin{aligned} L_{\mathrm{Poynt}}\propto & {} \,\rho _0\, v^{2.7} M^2 \,\,\, \mathrm{and} \end{aligned}$$
(36)
$$\begin{aligned} L_{\mathrm{Poynt, peak}}\approx & {} \,1.2\times 10^{46}\,\mathrm{erg\,s^{-1}} \left( \frac{\rho _0}{10^{-13}\,\mathrm{g\,cm^{-3}}}\right) \left( \frac{M}{10^8\,M_\odot }\right) ^2 \,\,, \end{aligned}$$
(37)
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.
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.
(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.
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.
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.
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.
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
$$\begin{aligned} L_{\mathrm{max}}=\frac{\varDelta E}{T}\approx 10\,\epsilon ^3 \frac{E_{\mathrm{bind}}}{P_{\mathrm{orb}}}\; . \end{aligned}$$
(38)
For example, \(\epsilon =0.05\) implies \(L_{\mathrm{max}}\approx 10^{-3} E_{\mathrm{bind}}/P_{\mathrm{orb}}\). The natural disk luminosity is
$$\begin{aligned} L_{\mathrm{disk}}\sim \frac{E_{\mathrm{bind}}}{P_{\mathrm{orb}}(r)/[3\pi \alpha (h/r)^2]} \sim 10^{-3}\, \frac{E_{\mathrm{bind}}}{P_{\mathrm{orb}}}\; , \end{aligned}$$
(39)
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 \(100\,M\), as suggested by Milosavljević and Phinney 2005).
If the circumbinary disk has followed the binary down to relativistic separations, then the situation is different. The primary reason is that circular orbits near the innermost stable circular orbit have specific angular momenta that do not vary much with radius, so a small change in the mass can lead to a large change in radius. For example, O’Neill et al. (2009) consider the effect of a fractional loss \(\epsilon \ll 1\) of the mass-energy of the central object on an annulus initially at radius \(r=xM_0\) in Schwarzschild coordinates. After the mass loss (assumed impulsive), the new radius is determined by
$$\begin{aligned} {x^2\over {x-3}}={x^{\prime 2}(1-\epsilon )^2\over {x^\prime -3}} \end{aligned}$$
(40)
where \(x^\prime \equiv r^\prime /M\), with \(r^\prime \) being the radius after circularization and \(M=M_0(1-\epsilon )\). The initial specific energy of the orbit is
$$\begin{aligned} E_{\mathrm{init}}=\sqrt{(x-2)[x-2(1-\epsilon )]\over {x(x-3)}} \end{aligned}$$
(41)
and the final specific energy after circularization is
$$\begin{aligned} E_{\mathrm{fin}}={x^\prime -2\over {\sqrt{x^\prime (x^\prime -3)}}}\; . \end{aligned}$$
(42)
The fractional change in the binding energy is then \(\varDelta E/E_{\mathrm{tot}}=(E_{\mathrm{init}}-E_{\mathrm{fin}})/(1-E_{\mathrm{init}})\) and the energy release of an annulus initially near the ISCO is much greater than it would be in Newtonian gravity. For example, for \(\epsilon =0.05\), the Newtonian fractional energy release is \(\varDelta E/E_{\mathrm{tot}}=0.0025\), but for an annulus initially at the ISCO of a Schwarzschild hole, \(x=6\), \(\varDelta E/E_{\mathrm{tot}}= 0.0679\), nearly thirty times larger. The spacetime will, of course, be dynamic and thus not Schwarzschild, but we expect the general principle of energy enhancement to apply.
Nonetheless, O’Neill et al. (2009) found that the prime result of mass loss is not an enhancement, but a deficit of emission. The reason is that after mass loss the inner edge of the disk circularizes at a radius larger (in gravitational units) that its original inner edge. Thus, until viscous effects can return the disk to its pre-merger radius, which can take several hundred M of time (amounting to days for a \(10^8\,M_\odot \) central black hole), the efficiency of emission and hence the luminosity will be less than it was pre-merger. Subsequent studies have confirmed this basic picture (Megevand et al. 2009; Corrales et al. 2010; Anderson et al. 2010; Rosotti et al. 2012; Zanotti 2012). Whether this effect is detectable depends on variables such as the primary band of emission and the extinction of the center of the host galaxy. It is also conceivable that the eventual filling in of the circumbinary gap would lead to the production of jets and thus radio emission where there was none previously, as discussed in Sect. 4.2.2. This would require that the source be face-on to us, which would occur in only a small fraction of all mergers, but it seems worth pursuing.
In Sect. 4.1 and 4.2 we discussed a number of signatures associated with the binary accretion flows that can provide the EM counterparts to their GW emission. In Table 1 we provide a summary of these signatures and their origins according to the wavelength band in which they are most likely to emerge. Most proposed EM counterparts involve variability on relatively short time scales and are expected to emerge in high-energy bands, indicating perhaps the need for a future X-ray timing mission that could carry out the EM followup.
Table 1 Summary of the EM counterparts to GW emission before, during and immediately after the MBH merger discussed in Sects. 4.1 and 4.2 In the next section we turn to signatures that might be visible months to millions of years after the merger, including kicks and afterglows.
Afterglows and other post-merger signatures
In addition to the coincident EM counterparts to MBH mergers that may be observable as near-simultaneous multimessenger events, EM afterglows could appear on longer (\(\sim \) months to decades) timescales after merger. The enormous release of energy and momentum during a MBH merger provides numerous possible mechanisms for perturbing the surrounding gas disk, if present. These sudden perturbations may induce oscillations, density waves, or shocks in the disk, all of which could produce some type of EM flare that is distinct from standard accretion emission. Here we discuss various physical mechanisms that may produce EM afterglows in the aftermath of MBH mergers, which could be crucial for localization of GW sources detected with LISA. We also discuss signatures of past MBH mergers that may form and persist on much longer timescales (thousands to millions of years), including signatures of GW-recoiling MBHs. Even though these long-timescale signatures cannot be associated with a specific GW observation, they could still place valuable constraints on the overall population and characteristics of merging MBHs.
Afterglows from circumbinary gap refilling
In Sect. 4.2.3 we discussed the immediate response of a gas disk to the MBH mass loss radiated away in GWs, which should produce a short-term decrease in the accretion luminosity as the disk retreats from the lower-mass MBH. After this initial phase, however, the disk will continue to refill the pre-merger circumbinary gap on a viscous timescale; completely refilling the gap will generally take months to years in the source rest frame, depending on the MBHB mass and decoupling radius. Throughout this gap refilling phase, a steady increase in the luminosity and photon energy of the emission is expected, as this inner portion of the accretion disk will radiate at higher energies with peak emission in the hard UV and X-ray bands (e.g., Milosavljević and Phinney 2005; Dotti et al. 2006; Tanaka and Menou 2010; Tanaka et al. 2010; Shapiro 2010). This scenario is therefore a candidate for producing an observable EM afterglow that could be seen in the months to years following a GW detection of a MBH merger.
Relatedly, Schnittman and Krolik (2008) argue that the large optical depth of the disk will thermalize energy from merger-induced perturbations, such that most of the afterglow emission should occur in the IR. For LISA sources with masses \(\sim 10^6\,M_{\odot }\), the timescale for this IR afterglow could be years to decades. This possibility is of particular interest because, unlike UV and soft X-ray emission, IR flares would be much less susceptible to attenuation by gas and dust.
If the accretion rate from the pre-merger circumbinary disk is near the Eddington luminosity prior to the decoupling of the MBHB from the disk, some estimates indicate that the gap-refilling afterglow could be quite luminous, perhaps even greater than the Eddington luminosity in extreme cases (Tanaka and Menou 2010; Shapiro 2010). However, the characteristics and detectability of this accretion flow are uncertain. For one, Tanaka et al. (2010) find that the afterglow luminosity is sensitive to the circumbinary disk properties, especially the surface density and the ratio of the viscous stress to gas pressure. More importantly, recent simulations of circumbinary disk evolution demonstrate that gas streams can readily flow through the circumbinary gap to fuel the mini-disks around individual MBHs at rates comparable to the accretion rate in a disk with no gap at all (e.g., D’Orazio et al. 2013; Farris et al. 2014; Shi and Krolik 2015; Tang et al. 2018; Bowen et al. 2017). The pileup of gas at the edge of the circumbinary gap predicted by earlier work may therefore be fairly minimal. While this greatly increases the possibilities for MBHBs to produce distinctive EM counterparts prior to merger, the continuous feeding of the mini-disks around each MBH would reduce the spectral and luminosity contrast between a post-merger EM flare and the emission from pre-merger accretion (e.g., Shi et al. 2012; Noble et al. 2012; Fontecilla et al. 2017; Bowen et al. 2018, 2019).
Afterglows from GW recoil kicks
As described in Sect. 3.4, asymmetric GW emission from an unequal-mass, misaligned, spinning MBHB merger can impart a recoil kick to the merged MBH of hundreds to thousands of km \(\hbox {s}^{-1}\). In the frame of the kicked MBH, this causes a velocity \({\mathbf {v}}_{\mathrm{kick}}\) to be added to the instantaneous velocity of the orbiting gas. The perturbation to the disk will induce strong density enhancements, which can shock the gas and produce luminous afterglows on timescales of weeks, months, or years after the MBH merger, depending on the disk and recoiling MBH properties (Fig. 29; Lippai et al. 2008; Schnittman and Krolik 2008; Megevand et al. 2009; Rossi et al. 2010; Corrales et al. 2010; Zanotti et al. 2010; Meliani et al. 2017).
Using a collisionless particle simulation, Lippai et al. (2008) demonstrated the formation of tightly-wound spiral density caustics resulting from a GW recoil kick oriented in the disk plane. In contrast, for a kick oriented perpendicularly to the disk, the resulting concentric density enhancements are weaker and delayed to potentially \(\gtrsim \) 1 year after the merger. This general picture of recoil-induced disk perturbations, and the dependence on kick direction, has been substantiated by later studies using full hydrodynamics simulations (Fig. 29; Megevand et al. 2009; Rossi et al. 2010; Corrales et al. 2010; Zanotti et al. 2010; Zanotti 2012; Ponce et al. 2012).
Rossi et al. (2010), Corrales et al. (2010), and Zanotti et al. (2010) all reach similar conclusions regarding the luminosity and timescale of a recoil-induced afterglow. They find that for a \(10^6 M_{\odot }\) MBH, the luminosity should rise steadily over a timescale of \(\sim \) months, reaching a peak of \(\sim \) a few \(\times 10^{43}\) erg \(\hbox {s}^{-1}\), or \(\sim 10\%\) of the Eddington luminosity at that MBH mass. This emission is likely to peak in the UV or soft X-ray bands. Variability is also a potential signature of these afterglows (Anderson et al. 2010; Rossi et al. 2010; Ponce et al. 2012).
A number of complicating factors may make these recoil afterglows difficult to detect in practice. Rossi et al. (2010) explored a range of recoil kick angles relative to the disk plane and concluded that the energy available for dissipation depends strongly on this angle, by up to three orders of magnitude. They also found that the observability of the afterglow depends on the mass of the disk at the radii where most energy is deposited – if the disk is truncated beyond the self-gravitating regime, then the afterglow luminosity is expected to be small, possibly undetectable.
The recoil velocity is an important factor as well: clearly, if all else is equal, higher-velocity recoils will produce stronger disk perturbations. The brightest afterglows would presumably result from high-velocity recoils oriented in the plane of the disk. However, the largest component of a high recoil velocity (arising from misaligned MBH spins) will generally be perpendicular to the binary orbital plane prior to merger. Contributions to the recoil velocity directed into the orbital plane are more likely to dominate in the case of aligned MBH spins, which produce much smaller kick speeds (e.g., Baker et al. 2008; van Meter et al. 2010b; Lousto et al. 2012). The MBH binary orbital plane need not be aligned with the plane of the circumbinary disk, of course, and it is reasonable to envision that such a scenario could occur in systems where the MBH spins remain misaligned with each other and with the disk all the way to decoupling.
Another possible source of recoil-induced EM flares is the fallback of marginally-bound gas after the recoiling MBH and its bound accretion disk leave the galaxy nucleus (Shields and Bonning 2008). This fallback could shock the surrounding gas, leading to a flare that should peak in the soft X-ray bands. The timescale for this type of flare (\(\sim 10^3\)–\(10^4\) yr) would not be accessible as an EM counterpart to a GW source on human timescales, but some of these could be detected in AGN surveys if they are sufficiently luminous. Shields and Bonning (2008) argue that the reprocessing of the soft X-ray flare by infalling material could produce a distinct UV/optical emission line spectrum.
Finally, attenuation by gas and dust is again likely to be a serious concern for observing many of these UV and soft X-ray afterglows, particularly because galactic nuclei often have high extinction. The possibility that some of this emission could be reprocessed and detected as an IR afterglow is therefore an appealing one to explore (Schnittman and Krolik 2008). We emphasize, however, that the detailed light curves and spectral evolution of any post-merger EM afterglows are still quite uncertain, so the possibility of observable flares at other wavelengths cannot be discounted.
Post-merger signatures of AGN jets
The morphology of AGN radio jets may contain signatures of a recent MBH merger. If a MBH has an associated radio jet prior to merging with another MBH, the rapid reorientation of its spin induced by the merger (“spin flip”) could create a second, younger radio jet that is misaligned with the older, pre-merger radio lobes (e.g., Merritt and Ekers 2002; Dennett-Thorpe et al. 2002). Liu (2004) suggest that similar features could be produced by the rapid reorientation of a BH by a misaligned accretion disk, via the Bardeen–Petterson effect. Such “X-shaped” radio sources are indeed observed in hundreds of AGNs (e.g., Murgia et al. 2001; Merritt and Ekers 2002; Liu 2004; Lal and Rao 2007; Roberts et al. 2015; Bera et al. 2020). Numerous studies have concluded that the morphology and spectral features of some of these sources favor a scenario in which BH spin reorientation occurred on a short timescale (\(\lesssim 10^5\) yr; e.g., Cheung 2007; Mezcua et al. 2011; Gopal-Krishna et al. 2012; Hernández-García et al. 2017; Saripalli and Roberts 2018). However, other studies find that these X-shaped radio sources are more likely to arise from physics that has nothing to do with MBH spins, such as the backflow of jet material after it collides with a dense intergalactic medium, or the expansion of a jet-inflated cocoon along the minor axis of an elliptical galaxy (e.g., Leahy and Williams 1984; Capetti et al. 2002; Kraft et al. 2005; Hodges-Kluck et al. 2010; Hodges-Kluck and Reynolds 2011; Rossi et al. 2017; Joshi et al. 2019; Cotton et al. 2020).
MBH mergers could also lead to the interruption of AGN jets, owing to either a drop in accretion following decoupling of a MBH binary from a circumbinary disk or following the MBH mass loss at merger. Liu et al. (2003) suggest that examples of these interrupted jets could be seen in double-double radio galaxies, where a younger set of radio jets is aligned with older radio lobes along the same axis. The inferred interruption timescales for many of these objects are quite long, however (\(\sim \) Myr). Alternately, if a jet (re)launches after the circumbinary disk responds to the merger, this event could be observed as an EM counterpart to a GW detection on timescales of weeks to years (e.g., Ravi 2018; Blecha et al. 2018; Yuan et al. 2021).
Long-lived accretion signatures of GW recoil
Another interesting possibility is the detection of GW recoiling MBHs long after the merger has occurred—up to millions or tens of millions years later. This scenario depends strongly on the pre-merger MBHB spins; perfectly-aligned spins will yield a maximum kick velocity \(< 200\) km \(\hbox {s}^{-1}\), while highly misaligned, rapidly spinning MBHs are necessary to produce kicks of up to 4000–5000 km \(\hbox {s}^{-1}\) (e.g., Campanelli et al. 2007a; Lousto et al. 2012). If MBHB spins are highly aligned prior to merger, the post-merger recoiling MBH will not stray far from the galactic center in most galaxies, and long-lived signatures of GW recoil will be difficult if not impossible to observe. As discussed in Sect. 3.2.2, MBHB spin evolution is poorly constrained, and the degree of spin misalignment is sensitive to the nature of the accretion flow and the degree to which the MBH binary environment is gas-dominated (e.g., King and Pringle 2006; Bogdanović et al. 2007; Lodato and Gerosa