ULTRAFAST OUTFLOWS FROM BLACK HOLE MERGERS WITH A MINIDISK

The discovery of gravitational waves (GWs) from GW 150914 by Advanced LIGO opens a new window of the high-energy universe (Abbott et al. 2016b). This first GW detection has simultaneously yielded the first observation of a binary black hole (BH) merger. The inferred initial masses ${36}_{-4}^{+5}{M}_{\odot }$ and ${29}_{-4}^{+4}{M}_{\odot }$ merged to form a final BH of mass ${62}_{-4}^{+4}{M}_{\odot }$ with the difference in final and initial masses corresponding to the energy emitted in GW radiation (Abbott et al. 2016b). This marks the beginning of GW astronomy (Abbott et al. 2016a) and offers a completely orthogonal means of observing the cosmos compared to the traditional avenues afforded by electromagnetically based telescopes.

It also heralds the beginning of a new era in multi-messenger astrophysics, in which both electromagnetic (EM), neutrino and GW probes are combined. For any class of GW sources, identifying EM and/or neutrino counterparts of the GW sources will enable us not only to study the dynamics and emission mechanisms of the transients, but also to obtain clues to environments where the sources are formed. EM-based telescopes have better localization capabilities than GW detectors, and strategic searches have been anticipated especially for binary systems involving a neutron star (NS) such as NS–NS and NS–BH mergers (e.g., Nissanke et al. 2013). On the other hand, an obvious EM counterpart is unexpected from BH–BH binary mergers in vacuum. As discussed in the context of supermassive BH binaries in the nucleus (Baruteau et al. 2012), expected EM signals depend on details of the setup. In other words, any counterpart would reveal the non-trivial presence of matter around BHs. For example, long-lived disks with small masses may be formed around BHs, which may lead to the possible existence of planets orbiting BHs (Perna et al. 2014). Also, it has been suggested that a BH binary in a hierarchal three-body system may resonantly trap a star, and its tidal disruption might emit EM signals around the coalescence of the two BHs (Seto & Muto 2011).

After the discovery of GW 150914, searches for EM counterparts of stellar-mass BH–BH mergers have become of more interest. A number of follow-up EM observations were indeed made from optical to gamma-ray energy bands (Evans et al. 2016; Fermi-LAT collaboration 2016; Savchenko et al. 2016; Smartt et al. 2016). A possible low-significance association with a short-duration gamma-ray burst (GRB) event has been reported with Fermi's GRB Monitor (GBM) just 0.4 s after GW 150914 (Connaughton et al. 2016). It is tentative and this signal has not been confirmed by INTEGRAL (Savchenko et al. 2016), but a number of possibilities to account for this have already been advanced (Li et al. 2016; Loeb 2016; Perna et al. 2016; Zhang 2016). A possible EM counterpart due to super-Eddington accretion onto the BH is discussed for BH–BH binaries embedded in active galaxies (Bartos et al. 2016; Stone et al. 2016). Afterglow emission of relativistic jets (Morsony et al. 2016; Yamazaki et al. 2016) has also been considered. Furthermore, the combination of EM and GW signals from the same source has been used to test the Einstein's equivalence principle (Wu et al. 2016) and modified dispersion relations for GWs (Collett & Bacon 2016).

In this work, we consider the fate of a possible minidisk accompanied by a BH binary. Our aim is to reveal consequences of such a system and to show that a possible EM counterpart signal can be used to test the proposed models. Our work does not rely on the tentative association with a short GRB. In Section 2 we consider fast optical transients that can emerge from disk-driven outflows. In Section 3 we study the possibility of long-lasting radio emission from blast waves originating from ultrafast outflows. In Section 4 we discuss a possible jet component. Throughout this work we use the notation $Q={10}^{x}{Q}_{x}$ in CGS unit unless noted otherwise.

If an accretion disk exists around a BH, a disk wind may be driven by radiation and/or magnetic fields in the disk corona. In particular, strong disk winds are commonly suggested in numerical studies of super-Eddington accretion disks (e.g., Ohsuga et al. 2005; Jiang et al. 2014; Sa̧dowski et al. 2014). Although the origin of the disk material is an open question, it has been suggested that a low-temperature "fossil" disk may exist around one of the BHs via the formation of the dead zone (Perna et al. 2014). This disk may become ionized and active around the coalescence of the BHs (Perna et al. 2016). The viscous time for the disk around a BH is

$$\begin{eqnarray}&&{t}_{{\rm{vis}}}=\displaystyle \frac{1}{\alpha {{\rm{\Omega }}}_{K}}{\left(\displaystyle \frac{{R}_{d}}{H}\right)}^{2}\simeq 1.4\;{\rm{s}}\;{M}_{{\rm{BH,1.78}}}^{-1/2}{R}_{d,8}^{3/2}{\alpha }_{-1}^{-1}{(h/0.3)}^{-2},\end{eqnarray} \tag{ 1 }$$

where α is the viscosity parameter, H is the disk scale height, ${{\rm{\Omega }}}_{K}=\sqrt{{{GM}}_{{\rm{BH}}}/{R}_{d}^{3}}$ is the Kepler rotation frequency, ${M}_{{\rm{BH}}}=60\;{M}_{\odot }\;{M}_{{\rm{BH,1.78}}}$ is the merged BH mass, and $h\equiv H/{R}_{d}$. Equation (1) may be applied once the magnetorotational instability (MRI) becomes effective. The MRI timescale is ${t}_{{\rm{MRI}}}\approx (4/3{{\rm{\Omega }}}_{K})\simeq 0.015\quad {\rm{s}}\quad {M}_{{\rm{BH,1.78}}}^{-1/2}{R}_{d,8}^{3/2}$, which is essentially the dynamical time. In this work, for illustrative purposes we take the BH mass to be ${M}_{{\rm{BH}}}\sim 10\mbox{--}100\;{M}_{\odot }$, a disk mass ${M}_{d}\sim {10}^{-5}-1\;{M}_{\odot }$ (which may be comparable to the Jupiter mass), and a disk size ${R}_{d}\sim {10}^{8}\mbox{--}{10}^{11}$ cm as reference parameters (e.g., Perna et al. 2016). One can easily consider cases of other BH-disk parameters. In the fossil disk model, the mass accretion rate is ${\dot{M}}_{d}\approx {M}_{d}/{t}_{{\rm{vis}}}\simeq 7.0\times {10}^{-4}$ ${M}_{\odot }\;{{\rm{s}}}^{-1}\;{M}_{d,-3}{M}_{{\rm{BH,1.78}}}^{1/2}{R}_{d,8}^{-3/2}{\alpha }_{-1}{(h/0.3)}^{2}$. Such a super-Eddington accretion event might also happen even before or after the merger depending on models. The post-merger violent accretion occurs if the disk remains cold by the merger time ${t}_{{\rm{GW}}}$ becomes shorter than ${t}_{{\rm{vis}}}$. On the other hand, the pre-merger accretion may occur if the disk is ionized for ${t}_{{\rm{GW}}}\gt {t}_{{\rm{vis}}}$. Alternatively, for a hierarchical three-body system, a star can be trapped resonantly (Seto & Muto 2011) and may lead to the tidal disruption. We expect that not only stars but also planets can be involved, and planet formation around a BH is also suggested (Perna et al. 2014). The tidal radius for a star with R_* and M_* is ${R}_{t}\approx {R}_{*}{({M}_{{\rm{BH}}}/{M}_{*})}^{1/3}$ $\simeq 2.5\ \times {10}^{11}\;{\rm{cm}}\quad {M}_{{\rm{BH,1.78}}}^{1/3}{M}_{*,0}^{-1/3}{R}_{*,10.8}$. The return time of the bound material is ${t}_{{\rm{td}}}\ \approx \pi {({R}_{*}^{3}{M}_{{\rm{BH}}}/2{{GM}}_{*}^{2})}^{1/2}$ $\simeq 2.4\times {10}^{4}\quad {\rm{s}}\quad {M}_{{\rm{BH,1.78}}}^{1/2}{M}_{*,0}^{-1}{R}_{*,10.8}^{3/2}$, leading to ${M}_{*}/(3{t}_{{\rm{td}}})\simeq 1.4\times {10}^{-5}\quad {M}_{\odot }\quad {{\rm{s}}}^{-1}\quad {M}_{{\rm{BH,1.78}}}^{-1/2}{M}_{*,0}^{2}{R}_{*,10.8}^{-3/2}$ (Evans & Kochanek 1989).

We parametrize the disk outflow rate as ${\dot{M}}_{w}={\eta }_{w}{\dot{M}}_{d}$, which is ${\dot{M}}_{w}\sim {10}^{-6}-{10}^{-3}\;{M}_{\odot }\;{{\rm{s}}}^{-1}$ (i.e., ${\eta }_{w}\sim 0.1\mbox{--}1$) as our typical parameters. The disk-wind velocity v_w is expected to be comparable to the escape velocity,

$$\begin{eqnarray}&&{v}_{{\rm{esc}}}={(2{{GM}}_{{\rm{BH}}}/{R}_{d})}^{1/2}\simeq 0.4\;c\;{M}_{{\rm{BH,1.78}}}^{1/2}{R}_{d,8}^{-1/2},\end{eqnarray} \tag{ 2 }$$

which can be a significant fraction of the speed of light c. Throughout this work, we assume a constant wind velocity (although radiative acceleration is possible). The density at the foot-point of the wind is so large that it is expected to be radiation-dominated (cf. Rossi & Begelman 2009; Strubbe & Quataert 2009; Kashiyama & Quataert 2015, for studies on tidal disruption events and massive stellar collapses). Its initial temperature is ${T}_{0}\simeq 1.3\times {10}^{9}\;{\rm{K}}\;{\dot{M}}_{w,-4}^{1/4}{M}_{{\rm{BH,1.78}}}^{1/8}{R}_{d,8}^{-5/8}$. The optical depth is defined by ${\tau }_{T}=\kappa \rho {v}_{w}t$ (where κ is the opacity). The Thomson optical depth at R_d is ${\tau }_{T}^{0}\simeq 4.5\times {10}^{9}\;{M}_{{\rm{BH,1.78}}}^{-1/2}{R}_{d,8}^{-1/2}{\dot{M}}_{w,-4}({\kappa }_{T}/0.34\;{{\rm{cm}}}^{2}\;{{\rm{g}}}^{-1})$, where ${\kappa }_{T}$ is the Thomson scattering opacity.

The disk wind can be regarded as a continuous outflow until ${r}_{w}\approx {v}_{w}{t}_{{\rm{acc}}}$, where ${t}_{{\rm{acc}}}$ is ${t}_{{\rm{vis}}}$ or ${t}_{{\rm{td}}}$. We have ${r}_{w}\simeq 1.7\times {10}^{10}\;{\rm{cm}}\;{R}_{d,8}{\alpha }_{-1}^{-1}{(h/0.3)}^{-2}$ in the fossil disk model and ${r}_{w}\simeq 9.1\times {10}^{12}\;{\rm{cm}}\;{R}_{d,11}^{-1/2}{M}_{{\rm{BH,1.78}}}{M}_{*,0}^{-1}{R}_{*,10.8}^{3/2}$ in the tidal disruption model, respectively. The temperature and density scale as $T\propto {r}^{-2/3}$ and $\rho \propto {r}^{-2}$, so that we expect ${\tau }_{T}\propto {r}^{-1}$ for $r\lt {r}_{w}$. The disk wind effectively ⁴ ceases at ${t}_{{\rm{acc}}}$, and for $r\gt {r}_{w}$ we may expect a homologous expansion of the outflow. Thereafter, the temperature and density scale as T ∝ r⁻¹ and $\rho \propto {r}^{-3}$, leading to ${\tau }_{T}\propto {r}^{-2}$. The Thomson optical depth at $r\gt {r}_{w}$ is ${\tau }_{T}\simeq 1.4\times {10}^{7}$ ${M}_{{\rm{BH,1.78}}}^{-1/2}$ ${R}_{d,8}^{1/2}{\dot{M}}_{w,-4}{r}_{w,10.5}^{-1}$ $({\kappa }_{T}/0.34\;{{\rm{cm}}}^{2}\;{{\rm{g}}}^{-1})$ ${(r/{r}_{w})}^{-2}$.

Initially, the photons are trapped in the outflow. But photons start to escape when the Thomson optical depth becomes

$$\begin{eqnarray}&&{\tau }_{T}^{{\rm{bo}}}\approx c/{v}_{w}\simeq 2.5\;{M}_{{\rm{BH,1.78}}}^{-1/2}{R}_{d,8}^{1/2}.\end{eqnarray} \tag{ 3 }$$

The condition ${\tau }_{T}({r}_{{\rm{bo}}})={\tau }_{T}^{{\rm{bo}}}$ gives the photon breakout radius

$$\begin{eqnarray}&&{r}_{{\rm{bo}}}\simeq 7.5\times {10}^{13}\;{\rm{cm}}\;{\dot{M}}_{w,-4}^{1/2}{r}_{w,10.5}^{1/2}{({\kappa }_{T}/0.34{{\rm{cm}}}^{2}{{\rm{g}}}^{-1})}^{1/2}.\end{eqnarray} \tag{ 4 }$$

Note that around this radius the flow expansion time is comparable to the photon diffusion time. The diffusion timescale at this radius is estimated to be

$$\begin{eqnarray}{t}_{{\rm{diff}}}^{{\rm{bo}}} & \approx & \displaystyle \frac{{\tau }_{T}^{{\rm{bo}}}{r}_{{\rm{bo}}}}{c}\simeq 6300\;{\rm{s}}\;{M}_{{\rm{BH,1.78}}}^{-1/2}{R}_{d,8}^{1/2}\\ & & \times {\dot{M}}_{w,-4}^{1/2}{r}_{w,10.5}^{1/2}{({\kappa }_{T}/0.34{{\rm{cm}}}^{2}{{\rm{g}}}^{-1})}^{1/2}.\end{eqnarray} \tag{ 5 }$$

After ${t}_{{\rm{diff}}}^{{\rm{bo}}}$, the thermal photons escape from the outflow and can be observed by optical telescopes. Indeed, the typical temperature of this thermal emission is

$$\begin{eqnarray}{T}_{{\rm{bo}}} & \approx & 1.1\times {10}^{4}\;{\rm{K}}\;{M}_{{\rm{BH,1.78}}}^{1/8}{R}_{d,8}^{1/24}\\ & & \times {\dot{M}}_{w,-4}^{-1/4}{r}_{w,10.5}^{-1/6}{({\kappa }_{T}/0.34{{\rm{cm}}}^{2}{{\rm{g}}}^{-1})}^{-1/2}.\end{eqnarray} \tag{ 6 }$$

Notably, the value is quite insensitive to various parameters such as ${M}_{{\rm{BH}}}$, R_d, ${\dot{M}}_{w}$, and r_w. Thus, the predictions about these fast optical transients are promising as long as a minidisk exists in a BH–BH merger. The peak bolometric luminosity is estimated to be

$$\begin{eqnarray}{L}_{{\rm{th}}}^{{\rm{pk}}} & \approx & \displaystyle \frac{4\pi {r}_{{\rm{bo}}}^{3}{{aT}}_{{\rm{bo}}}^{4}}{{3t}_{{\rm{diff}}}^{{\rm{bo}}}}\simeq 3.6\times {10}^{40}\;{\rm{erg}}\;{{\rm{s}}}^{-1}\;{M}_{{\rm{BH,1.78}}}{R}_{d,8}^{-1/3}\\ & & \times {r}_{w,10.5}^{1/3}{({\kappa }_{T}/0.34{{\rm{cm}}}^{2}{{\rm{g}}}^{-1})}^{-1}.\end{eqnarray} \tag{ 7 }$$

For the above nominal values, the bolometric flux is ${F}_{{\rm{bol}}}\ ={L}_{{\rm{bol}}}/(4\pi {d}^{2})\sim 2.9\quad \times \quad {10}^{-14}\quad {d}_{26.5}^{-2}$ erg cm⁻² s⁻¹ (where d is the distance to the source); the spectral peak ${\nu }_{{\rm{pk}}}=2.82{k}_{B}{T}_{{\rm{bo}}}/h\simeq 6.5\times {10}^{14}$ Hz is in the B- or V-band, and the spectral flux ${F}_{\nu }\sim 4.4\times {10}^{-29}$ erg cm⁻² s⁻¹ Hz${}^{-1}=4.4$ μJy corresponds to a magnitude $m\simeq 22+5\mathrm{log}({d}_{26.5})$. Note also that the bolometric luminosity is proportional to ${M}_{{\rm{BH}}}$. Thus, it is easier to see the EM counterparts of BH mergers involving more massive BHs. Knowing the detailed shape of the BH mass function in BH–BH mergers, which depends on formation scenarios (e.g., Kinugawa et al. 2014; Abbott et al. 2016a; Belczynski et al. 2016; O'Leary et al. 2016), would be relevant to predict the detection rate of the fast optical transients.

Until the outflow reaches the photospheric radius, the bolometric luminosity is roughly constant, i.e., ${L}_{{\rm{th}}}^{{\rm{pk}}}\propto {t}^{0}$. The photospheric radius is

$$\begin{eqnarray}{r}_{{\rm{ph}}} & \approx & 1.2\times {10}^{14}\;{\rm{cm}}\;{M}_{{\rm{BH,1.78}}}^{-1/4}{R}_{d,8}^{1/4}\\ & & \times {\dot{M}}_{w,-4}^{1/2}{r}_{w,10.5}^{1/2}{({\kappa }_{T}/0.34{{\rm{cm}}}^{2}{{\rm{g}}}^{-1})}^{1/2},\end{eqnarray} \tag{ 8 }$$

which is reached at the time

$$\begin{eqnarray}{t}_{{\rm{ph}}} & \approx & {r}_{{\rm{ph}}}/{v}_{w}\simeq 9900\;{\rm{s}}\;{M}_{{\rm{BH,1.78}}}^{-3/4}{R}_{d,8}^{3/4}\\ & & \times {\dot{M}}_{w,-4}^{1/2}{r}_{w,10.5}^{1/2}{({\kappa }_{T}/0.34{{\rm{cm}}}^{2}{{\rm{g}}}^{-1})}^{1/2}.\end{eqnarray} \tag{ 9 }$$

The bolometric luminosity is thereafter expected to rapidly drop as ${L}_{{\rm{th}}}^{{\rm{pk}}}\propto {t}^{-2}$ just after ${t}_{{\rm{ph}}}$ and shows the exponential decay at $t\gg {t}_{{\rm{ph}}}$.

As shown above, the duration of the expected thermal emission is rather short, lasting from hours to days, which also lies at the frontier of optical surveys (Kulkarni 2012). Typical surveys in the present day (e.g., Pan-STARRS) may achieve a photometric magnitude of $m\sim 20\mbox{--}22$ that is hard to see the event at $d\sim 400$ Mpc, but nearby post-merger emission could be seen. Future LSST (with $m\sim 24.5$) is more promising. There may be possible confusions with optical transients from, e.g., super-Eddington outbursts of Galactic X-ray binaries (e.g., Revnivtsev et al. 2002), but they are expected to show persistent emission that can be distinguished. Even though it is challenging to make a follow-up observation, detecting such short optical transients would enable us to unequivocally identify the EM counterparts of BH–BH mergers. Blind searches with optical monitors with a wide field of view would be relevant for testing the model. The localization of the BH–BH mergers would in turn allow us to study their host galaxies, environments, and formation mechanisms.

An ultrafast flow originating from a minidisk wind develops into a blast wave, which starts to slow down at the deceleration radius

$$\begin{eqnarray}&&{r}_{{\rm{dec}}}\approx {\left(\displaystyle \frac{3{M}_{w}}{4\pi {{nm}}_{p}}\right)}^{1/3}\simeq 3.1\times {10}^{17}\;{\rm{cm}}\;{M}_{w,-4}^{1/3}{n}^{-1/3},\end{eqnarray} \tag{ 10 }$$

which is essentially the Sedov radius. Here n is the ambient density. The corresponding deceleration time is

$$\begin{eqnarray}&&{t}_{{\rm{dec}}}\approx {r}_{{\rm{dec}}}/{v}_{w}\simeq 2.5\times {10}^{7}\;{\rm{s}}\;{M}_{{\rm{BH,1.78}}}^{-1/2}{R}_{d,8}^{1/2}{M}_{w,-4}^{1/3}{n}^{-1/3}.\end{eqnarray} \tag{ 11 }$$

The shock velocity at $t\gt {t}_{{\rm{dec}}}$ is $v\approx 0.4{v}_{w}{(t/{t}_{{\rm{dec}}})}^{-3/5}\;$ $\simeq \;6.1\ \times {10}^{9}\;{\rm{cm}}\;{{\rm{s}}}^{-1}\;{M}_{{\rm{BH,1.78}}}^{1/5}$ ${R}_{d,8}^{-1/5}{M}_{w,-4}^{1/5}{n}^{-1/5}{t}_{7.5}^{-3/5}$. Assuming an adiabatic index $\hat{\gamma }=5/3$, the post-shock magnetic field is estimated to be $B={(9\pi {\epsilon }_{B}{{nm}}_{p}{v}^{2})}^{1/2}$ $\simeq 4.2\;{\rm{mG}}\quad {M}_{{\rm{BH,1.78}}}^{1/5}{R}_{d,8}^{-1/5}{M}_{w,-4}^{1/5}{n}^{3/10}{\epsilon }_{B,-2}^{1/2}{t}_{7.5}^{-3/5}$, where ${\epsilon }_{B}$ is the energy fraction carried by magnetic fields compared with the downstream thermal energy density. We take ${\epsilon }_{B}\sim 0.01$ as used in the literature of GRBs and trans-relativistic supernovae (Mészáros 2006).

We expect that electrons are accelerated at the forward shock, which leads to broadband synchrotron emission (cf. Mészáros & Rees 1993, for GRBs). The injection Lorentz factor of electrons at a non-relativistic shock is given by

$$\begin{eqnarray}{\gamma }_{{ei}} & \approx & ({\zeta }_{e}/2)({m}_{p}/{m}_{e})({v}^{2}/{c}^{2})\\ & & \simeq 7.2\;{M}_{{\rm{BH,1.78}}}^{2/5}{R}_{d,8}^{-2/5}{M}_{w,-4}^{2/5}{n}^{-2/5}({\zeta }_{e}/0.4){t}_{7.5}^{-6/5},\end{eqnarray} \tag{ 12 }$$

where ${\zeta }_{e}$ is a numerical coefficient related the energy fraction and injection fraction of accelerated electrons. We adopt ${\zeta }_{e}\sim 0.4$ based on the recent results of particle-in-cell simulations (Park et al. 2015). The accelerated electrons cool mainly via synchrotron radiation on a timescale ${t}_{{\rm{syn}}}\approx 6\pi {m}_{e}c/({\sigma }_{T}{B}^{2}{\gamma }_{e})$. From the condition ${t}_{{\rm{syn}}}=t$, the cooling Lorentz factor of the electrons is estimated to be

$$\begin{eqnarray}{\gamma }_{{ec}} & \approx & \displaystyle \frac{6\pi {m}_{e}c}{{\sigma }_{T}{B}^{2}t}\\ & & \simeq 2.9\times {10}^{6}\;{M}_{{\rm{BH,1.78}}}^{-2/5}{R}_{d,8}^{2/5}{M}_{w,-4}^{-2/5}{n}^{-3/5}{\epsilon }_{B,-2}^{-1}{t}_{7.5}^{1/5}.\end{eqnarray} \tag{ 13 }$$

The acceleration time of electrons via diffusive shock acceleration is given by ${t}_{{\rm{acc}}}\approx (20/3)c{\gamma }_{e}{m}_{e}{c}^{2}/({{eBv}}^{2})$ in the Bohm limit. The condition ${t}_{{\rm{acc}}}={t}_{{\rm{syn}}}$ gives the maximum Lorentz factor of electrons, which is

$$\begin{eqnarray}{\gamma }_{{eM}} & \approx & {\left(\displaystyle \frac{9\pi {{ev}}^{2}}{10{\sigma }_{T}{{Bc}}^{2}}\right)}^{1/2}\simeq 1.2\times {10}^{8}\;{M}_{{\rm{BH,1.78}}}^{1/10}{R}_{d,8}^{-1/10}\\ & & \times {M}_{w,-4}^{1/10}{n}^{-7/20}{\epsilon }_{B,-2}^{-1/4}{t}_{7.5}^{-3/10}.\end{eqnarray} \tag{ 14 }$$

With the above parameters, synchrotron spectra can now be calculated. Because we have ${\gamma }_{{ei}}\ll {\gamma }_{{ec}}$, the resulting spectrum is expected in the slow-cooling regime. The injection synchrotron frequency ${\nu }_{i}$ is given by

$$\begin{eqnarray}{\nu }_{i} & \approx & {\gamma }_{{ei}}^{2}\displaystyle \frac{3{eB}}{4\pi {m}_{e}c}\simeq 6.4\times {10}^{5}\;{\rm{Hz}}\;{M}_{{\rm{BH,1.78}}}{R}_{d,8}^{-1}\\ & & \times {M}_{w,-4}{n}^{-1/2}{\epsilon }_{B,-2}^{1/2}{({\zeta }_{e}/0.4)}^{2}{t}_{7.5}^{-3},\end{eqnarray} \tag{ 15 }$$

the cooling synchrotron frequency ${\nu }_{c}$ is

$$\begin{eqnarray}&&{\nu }_{c}\simeq 1.0\times {10}^{17}\;{\rm{Hz}}\;{M}_{{\rm{BH,1.78}}}^{-3/5}{R}_{d,8}^{3/5}{M}_{w,-4}^{-3/5}{n}^{-9/10}{\epsilon }_{B,-2}^{-3/2}{t}_{7.5}^{-1/5},\end{eqnarray} \tag{ 16 }$$

and the maximum synchrotron frequency is

$$\begin{eqnarray}&&{\nu }_{M}\simeq 1.7\times {10}^{20}\;{\rm{Hz}}\;{M}_{{\rm{BH,1.78}}}^{2/5}{R}_{d,8}^{-2/5}{M}_{w,-4}^{2/5}{n}^{-2/5}{t}_{7.5}^{-6/5}.\end{eqnarray} \tag{ 17 }$$

The peak synchrotron flux, which occurs at ${\nu }_{i}$ in the slow-cooling case, is

$$\begin{eqnarray}{F}_{\nu }^{{\rm{max}}} & \approx & \displaystyle \frac{0.6{f}_{e}{{nr}}^{3}}{4\pi {d}^{2}}\displaystyle \frac{4\sqrt{3}\pi {e}^{3}B}{3{m}_{e}{c}^{2}}\\ & \simeq & 5.0\;{\rm{mJy}}\;{M}_{{\rm{BH,1.78}}}^{4/5}{R}_{d,8}^{-4/5}{M}_{w,-4}^{4/5}\\ & & \times {n}^{7/10}{f}_{e}{\epsilon }_{B,-2}^{1/2}{t}_{7.5}^{3/5}{d}_{26.5}^{-2},\end{eqnarray} \tag{ 18 }$$

where f_e is the number fraction of accelerated electrons. The synchrotron spectrum at ${\nu }_{i}\lt \nu \lt {\nu }_{c}$ is ${F}_{\nu }\propto {\nu }^{1/2-s/2}$, where s is the injection spectral index of the accelerated electrons (which is defined by ${{dN}}_{e}/d{\gamma }_{e}\propto {\gamma }_{e}^{-s}$). The spectrum becomes ${F}_{\nu }\propto {\nu }^{-s/2}$ at ${\nu }_{c}\lt \nu \lt {\nu }_{M}$. The radio and optical band typically lies in the range of ${\nu }_{i}\lt \nu \lt {\nu }_{c}$, where the synchrotron flux at time t is approximately given by

$$\begin{eqnarray}{F}_{\nu } & \sim & 0.03\;{\rm{mJy}}\;{\nu }_{9}^{1/2-s/2}{M}_{{\rm{BH,1.78}}}^{3/10+s/2}{R}_{d,8}^{-3/10-s/2}{M}_{w,-4}^{3/10+s/2}\\ & & \times {n}^{19/10-s/4}{f}_{e}{\epsilon }_{B,-2}^{1/4+s/4}{({\zeta }_{e}/0.4)}^{-1+s}{t}_{7.5}^{21/10-3\;s/2}{d}_{26.5}^{-2}.\end{eqnarray} \tag{ 19 }$$

The detection of non-thermal radio signals is possible for nearby BH–BH mergers unless the ambient number density is too small. For comparison, the Very Large Array has a sensitivity of ∼0.03–0.1 mJy. An advantage of the synchrotron radio signals is that the emission is long-lasting, so that follow-up observations can be made on the scale of months to years after the detections of GW signals by Advanced LIGO, Advanced VIRGO, and KAGRA (Somiya 2012).

In principle, the post-merger emission from a relativistic jet could also be expected for the same BH-disk system. In particular, if the association with a short-duration GRB detected by Fermi-GBM is real, such a jet component is necessary to explain the observed gamma-ray luminosity of ${L}_{\gamma }^{{\rm{iso}}}\sim {10}^{49}\;{\rm{erg}}\;{{\rm{s}}}^{-1}$. As commonly discussed in the literature of EM counterparts of supermassive BH binaries (Mösta et al. 2010; Palenzuela et al. 2010; Schnittman 2011), a merged BH is spinning and its rotation energy can be extracted via the Blandford–Znajek (BZ) process (Blandford & Znajek 1977). The absolute jet luminosity is limited by (Blandford & Znajek 1977; McKinney 2005; Tchekhovskoy et al. 2011)

$$\begin{eqnarray}{L}_{{\rm{BZ}}} & \approx & \displaystyle \frac{1}{6c}{{\rm{\Omega }}}_{H}^{2}{B}_{p}^{2}{R}_{H}^{4}\\ & \simeq & 1.27\times {10}^{47}\;{\rm{erg}}\;{{\rm{s}}}^{-1}\;{B}_{p,12}^{2}{M}_{{\rm{BH,1.78}}}^{2},\end{eqnarray} \tag{ 20 }$$

for the dimensionless Kerr parameter of $a\sim 0.7$. Here ${{\rm{\Omega }}}_{H}$ is the BH rotation frequency, R_H is the horizon radius, and B_p is the magnetic field anchored to the BH. The magnetic field would be supplied by the minidisk via magnetohydrodynamic instabilities such as the MRI, although the formation of ordered magnetic fields in the BH magnetosphere is uncertain. But a rough upper limit on B_p can be placed by ${B}_{p}^{2}/(8\pi )\lesssim {\dot{M}}_{d}c/(4\pi {R}_{H}^{2})$, which leads to ${B}_{p}\lesssim 2.1\ \times {10}^{13}\;{\rm{G}}\;{\dot{M}}_{d,-3}^{1/2}{M}_{{\rm{BH,1.78}}}^{-1}$, although the maximum value of B_p seems to require extreme conditions. Note that the BZ outflow may be significantly collimated, where the isotropic-equivalent luminosity is enhanced by the inverse of the beaming factor $2{\theta }_{j}^{-2}$, where ${\theta }_{j}$ is the jet opening angle. In the fossil disk model, ${t}_{{\rm{acc}}}\approx {t}_{{\rm{vis}}}$ is so short that the jet emission can be more luminous than Galactic X-ray binaries, even if the jet launching mechanism may be the same.

The jet luminosity is often expressed as ${L}_{{\rm{BZ}}}={\eta }_{j}{\dot{M}}_{d}{c}^{2}$. For ${\dot{M}}_{d}\sim {10}^{-3}\;{M}_{\odot }\;{{\rm{s}}}^{-1}$, ${\eta }_{j}$ is expected to be $\sim {10}^{-4}-{10}^{-3}$ for ${B}_{p}\sim {10}^{12}\mbox{--}{10}^{13}\;{\rm{G}}$. Noting that the disk-wind luminosity is ${L}_{w}={\eta }_{w}{\dot{M}}_{d}{v}_{w}^{2}\sim {\eta }_{w}{\dot{M}}_{d}{c}^{2}$, we obtain ${L}_{{\rm{BZ}}}/{L}_{w}\ \sim {10}^{-2}{\eta }_{j,-3}{\eta }_{w,-1}^{-1}$. If the BZ outflow is collimated, the jet component can overwhelm the total flux received by an on-axis observer. Accordingly, the afterglow radio emission would be dominated by the jet component (Morsony et al. 2016; Yamazaki et al. 2016). However, note that the wind component is still expected to be significant for super-Eddington accretion. In particular, the wind emission would be a dominant component for off-axis observers, which is relevant in the search for EM counterparts of a bulk of GW sources.

An interesting question of BH–BH mergers is whether they can be potential cosmic-ray accelerators and associated neutrino sources. The isotropic-equivalent magnetic luminosity that is required to accelerate cosmic rays with energy E is (Blandford 2000; Waxman 2004)

$$\begin{eqnarray}&&{L}_{B}^{{\rm{iso}}}\gtrsim \displaystyle \frac{1}{2}\displaystyle \frac{{E}^{2}}{{Z}^{2}{e}^{2}}{{\rm{\Gamma }}}^{2}c,\end{eqnarray} \tag{ 21 }$$

where ${L}_{B}^{{\rm{iso}}}$ is the isotropic-equivalent magnetic luminosity, Γ is the Lorentz factor of the acceleration region, and Z is the charge of the cosmic-ray particles. Noting that causality implies the condition ${\rm{\Gamma }}{\theta }_{j}\gtrsim 1$, we have

$$\begin{eqnarray}&&E\lesssim {\left(\displaystyle \frac{4{Z}^{2}{e}^{2}{L}_{{\rm{BZ}}}}{{{\rm{\Gamma }}}^{2}{\theta }_{j}^{2}c}\right)}^{1/2}\lesssim 2.6\times {10}^{22}\;{\rm{eV}}\;Z{\dot{M}}_{d,-3}^{1/2}.\end{eqnarray} \tag{ 22 }$$

Although the maximum value indicated above would be too extreme, BH–BH mergers could be potential accelerators of cosmic rays. Correspondingly, they could be sources of high-energy neutrinos as well (cf. Thompson & Lacki 2011, for supermassive BH binaries), although all predictions depend on details of the dissipation and emission mechanisms. Interestingly, the observed BH merger rate is not far from the short GRB rate. However, even if the cosmic-ray energy per BH merger reaches ${{ \mathcal E }}_{{\rm{cr}}}^{{\rm{iso}}}\sim {10}^{50}\;{\rm{erg}}$, the luminosity density ${{ \mathcal E }}_{{\rm{cr}}}^{{\rm{iso}}}{\rho }_{{\rm{dBH}}}\ ={10}^{42}\;{\rm{erg}}\;{{\rm{Mpc}}}^{-3}\;{{\rm{yr}}}^{-1}$ $({{ \mathcal E }}_{{\rm{cr}}}^{{\rm{iso}}}/{10}^{50}\;{\rm{erg}})$ $({\rho }_{{\rm{dBH}}}/10\;{{\rm{Gpc}}}^{-3}\;{{\rm{yr}}}^{-1})$ is far below the energy budget of ultra-high-energy cosmic rays, ${10}^{44}\;{\rm{erg}}\;{{\rm{Mpc}}}^{-3}\;{{\rm{yr}}}^{-1}$ (e.g., Murase & Takami 2009).

We considered the fate of ultrafast disk winds from possible minidisks associated with stellar-mass BH–BH mergers. We have shown that: (1) fast disk winds will lead to fast optical transients that shine in a timescale from hours to days; and (2) the outflows interacting with the interstellar medium will cause a strong forward shock and synchrotron emission from the accelerated electrons may expected on a timescale of years. The identifications of such EM counterparts are of interest independently of whether the possible association with short GRBs is real. While the detections may be challenging, the existing models involving disks can be tested by dedicated follow-up observations and/or monitoring searches.

We also discussed a possible physical origin for a tentative short GRB as reported by Connaughton et al. (2016). The jet powered by the BZ process could potentially give a viable explanation for the observed luminosity in terms of a relativistic jet component, if the disk exists.

The authors acknowledge support by the Pennsylvania State University (K.M. and I.S.), the NASA Einstein Fellowship program (K.K.), and NASA NNX13AH50G (P.M. and N.S.).