From the scattering experiments, we have obtained the HVB distribution d3Nhvb/(dmchirpdvejdτ). Note that this distribution varies with M1 and q, thus it is a function of mchirp, τ and veject but also of log M1 and log q. After the ejection, a given HVB will escape from the galactic potential with initial velocity vej and merge in the dark matter halo or in the IGM at a distance R away from the host galactic centre. This distance depends on the flight time τ, and can be computed explicitly by assuming a specific galactic potential. In this work, we adopt the characteristic galactic potential of elliptical galaxies (e.g. Kenyon et al. 2008, 2014),
(19)$$\begin{eqnarray*}
\phi (r)&amp; =&amp; \phi _\mathrm{bh}(r)+\phi _b(r) + \phi _h(r)\nonumber\\
&amp;=&amp; -\frac{G(M_1+M_2)}{r} -\frac{GM_\mathrm{ b}}{r+r_\mathrm{ b}} -\frac{GM_\mathrm{ h}}{r}\ln \left(1+\frac{r}{r_\mathrm{ h}}\right),
\end{eqnarray*}$$where Mb is the mass of the bulge, Mh is the mass of the dark matter halo, rb is the scale radius of the bulge, and rh is the scale radius of the dark matter halo. The choice of the spherical potential in equation (19) is dictated by computational convenience, as it allows us to perform 1D calculations to compute the ejected CO binary trajectory. The addition of a disc component would produce a deflection of the ejected trajectories, without significantly affecting the overall distribution of distances travelled. We scale the variables Mb, Mh, rb, and rh with the total mass of the SMBHB M1(1 + q) by means of the relations (e.g. Kormendy & Ho 2013; Kulier & Ostriker 2015)
(20)$$\begin{eqnarray*}
\frac{M_1(q+1)}{10^9\mathrm{ M}\mathrm{ }_\odot } &amp;=&amp; 0.49\left(\frac{M_b}{10^{11}\mathrm{ M}_\odot }\right)^{1.16},\nonumber\\
\frac{M_h}{10^{13}\mathrm{ M}_\odot } &amp; = &amp; 0.30\left(\frac{M_b}{10^{11}\mathrm{ M}_\odot}\right),\nonumber\\
\frac{M_b}{10^6 \mathrm{ M}_\odot }\bigg (\frac{\mathrm{kpc}}{r_b}\bigg)^3 &amp;=&amp; 4000, \nonumber\\
\frac{M_h}{10^6 \mathrm{ M}_\odot }\bigg (\frac{ \mathrm{kpc}}{r_h}\bigg)^3 &amp;=&amp; 125.
\end{eqnarray*}$$The ejection properties are recorded at a distance of $r\, \sim \, 50a_{\mathrm{bh}}$, given the condition that the total energy of the compact binary must be positive. Thus, we can calculate the travelled distance R = D(M1(1 + q), τ, vej)
(21)$$\begin{eqnarray*}
\int _0^D\frac{\mathrm{ d}r}{\sqrt{2E_0-2\phi (r)- 2l_0^2/r^2}} =\int _0^{\tau _\mathrm{GW}} \mathrm{ d}t,
\end{eqnarray*}$$where E0 and l0 are, respectively, the total energy and total angular momentum per unit mass at the time of ejection. The distribution of HVBs as a function of mass, travelling time τ, and distance from the galaxy centre at a merger distance R can then be calculated as
(22)$$\begin{eqnarray*}
&amp;&amp; {\frac{\mathrm{d}^3 N_\mathrm{hvb}}{\mathrm{d}m_\mathrm{chirp}\mathrm{d}\tau \mathrm{d}R}(\log q,\log M_1,m_\mathrm{chirp},\tau ,R)}\nonumber\\
&amp;&amp; {\quad=\frac{\mathrm{d}}{\mathrm{d}R}\oint _{D(M_1,q,\tau ,v_\mathrm{ej}) \lt R}\frac{\mathrm{d}^3 N_\mathrm{hvb}}{\mathrm{d}m_\mathrm{chirp}\mathrm{d}v_\mathrm{ej}\mathrm{d}\tau }\mathrm{d}v_\mathrm{ej}.}
\end{eqnarray*}$$Fig. 10 shows the 2D-PDF of the distance R and the corresponding velocity at that distance of the simulated HVBs. Note that not all ejected mergers can be observed within the age of the universe.
Citation: Kormendy & Ho 2013
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