Formation rate of extreme mass ratio inspirals in active galactic nuclei

Extreme mass ratio inspirals (EMRIs) are important sources for space-borne gravitational wave detectors, such as the laser interferometer space antenna and TianQin. Previous EMRI rate studies have focused on the “loss cone” scenario, where stellar-mass black holes (sBHs) are scattered into highly eccentric orbits near the central massive black hole (MBH) via multibody interaction. In this work, we calculate the rate of EMRIs of an alternative formation channel: EMRI formation assisted by the accretion flow around accreting massive black holes. In this scenario, sBHs and stars on inclined orbits are captured by the accretion disk, and then subsequently migrate towards the MBH, under the influence of density wave generation and head wind. By solving the Fokker-Planck equation incorporating both sBH-sBH–sBH-star scatterings and sBH–star-disk interactions, we find that an accretion disk usually boosts the EMRI formation rate per individual MBH by $\mathcal{O}({10}^1–{10}^3)$ compared with the canonical loss cone formation channel. Taking into account that the fraction of active Galactic nuclei (AGNs) is $\sim \mathcal{O}({10}^{-2}-{10}^{-1})$, where the MBHs are expected to be rapidly accreting, we expect EMRI formation assisted by AGN disks to be an important channel for all EMRIs observed by space-borne gravitational wave detectors. These two channels also predict distinct distributions of EMRI eccentricities and orbit inclinations with respect to the MBH spin equatorial plane, which can be tested by future gravitational wave observations.

Figure 1

Final distribution functions $f_{\mathrm{star}}(t_{\mathrm{f}},E,R)$ (left panel), $f_{\mathrm{bh}}(t_{\mathrm{f}},E,R)$ (middle panel). Initial and final $R$-integrated distribution functions (right panel). All the distribution functions are shown in units of ${10}^5{\mathrm{pc}}^{-3}/(2\pi {\sigma }^2{)}^{3/2}$ and energies $E$ are shown in units of ${\sigma }^2$.

Figure 2

Mass segregation in the $R$ direction, where the heavier component is more concentrated on circular orbits.

Figure 3

The time dependence of the inflow rate ${\overline{F}}_E{|}_{E=E_{\mathrm{gw}}}$, the EMRI rate ${\mathrm{\Gamma }}_{\mathrm{emri}}$, and the sBH number growth rate $d{N}_{\mathrm{bh}}(E>E_{\mathrm{gw}})/dt$.

Figure 4

Upper panel: The dependence of the characteristic EMRI rate per MBH ${\widehat{\mathrm{\Gamma }}}_{\mathrm{emri}}$ on the MBH mass $M_{•}$ for different initial cluster models, where $(\delta ={\delta }_0={10}^{-3},\gamma =1.5)$ is our fiducial model, $(\delta =2{\delta }_0,\gamma =1.5)$ is a similar cluster model with higher sBH fraction, and $(\delta ={\delta }_0,\gamma =1.8)$ is a Galactic nuclear stellar clusterlike model. Lower panel: the average rate ${\overline{\mathrm{\Gamma }}}_{\mathrm{emri}}$.

Figure 5

Fiducial $\alpha /\beta$ disk in the upper and lower row. Left panel: disk structure with $\mathrm{\Sigma }[{\mathrm{g}/\mathrm{cm}}^3]$ the surface density, $T[\mathrm{eV}]$ the middle plane temperature, $\tau$ the disk optical depth, and $h≔H/r$ the disk aspect ratio. Middle panel: the torques (in units of $c^2=1$) exerted on the sBH from GW emission ${\dot{J}}_{\mathrm{gw}}$ [(40)], disk wind ${\dot{J}}_{\mathrm{wind}}$ [(36)] and density waves ${\dot{J}}_{\mathrm{mig},\mathrm{I}}$ [(30)], where ${\dot{J}}_{\mathrm{mig},\mathrm{I}}$ changes its sign close to the local density maxima. Right panel: the corresponding timescales (in units of yr) on which different torque change the sBH orbital angular momentum by order of unity $t_i≔J/|{\dot{J}}_i|$. with $i=\text{gw}/\mathrm{wind}/\mathrm{mig},\mathrm{I}$. For comparison, the typical disk life span $\sim ({10}^6,{10}^8)\mathrm{yr}$ is plotted as a horizontal gray band.

Figure 6

Same as Fig. 5 except for a fiducial TQM disk.

Figure 7

Upper row: Same as Fig. 5 except for a comparison of the $\alpha$ disk with low viscosity $\alpha =0.01$, where the gap opening condition is satisfied when an orbiting sBH is located in the range of $\sim ({10}^3,{10}^4)M_{•}$ (vertical gray band in the middle panel). Lower row: Same to Fig. 5 except for a comparison $\alpha$ disk with $M_{•}=4\times {10}^8{M}_{\odot }$ and ${\dot{M}}_{•}=0.5{\dot{M}}_{•}^{\text{Edd}}$, where the type-I torque becomes positive in two disconnected regions.

Figure 8

The dependence of $E_{\mathrm{crit}}[{\sigma }^2]$ on the disk lifetime $T_{\mathrm{disk}}[\mathrm{yr}]$ for the fiducial $\alpha$ disk shown in Fig. 5.

Figure 9

We show the distribution functions for the case of fast disk capture (${\mu }_{\mathrm{cap}}=1$) in the upper row: $f_{\mathrm{star}}(t={10}^7\mathrm{yr},E,R)$ (left panel), $f_{\mathrm{bh}}(t={10}^7\mathrm{yr},E,R)$ (middle panel), the time dependence of EMRI rate ${\mathrm{\Gamma }}_{\mathrm{emri}}$, where $f_i$ are shown in units of ${10}^5{\mathrm{pc}}^{-3}/(2\pi {\sigma }^2{)}^{3/2}$ and $E$ is shown in units of ${\sigma }^2$. The counterparts of the slow disk capture (${\mu }_{\mathrm{cap}}=0.1$) case are in the lower row.

Figure 10

The time dependence of disk assisted EMRI rate per MBH ${\mathrm{\Gamma }}_{\mathrm{emri}}(t;T_{\mathrm{disk}})$ for different disk lifetimes $T_{\mathrm{disk}}$ and different disk capture parameters ${\mu }_{\mathrm{cap}}$. The dashed line is the average EMRI rate via loss cone of the fiducial model.

Figure 11

The disk-assisted EMRI rate ${\mathrm{\Gamma }}_{\mathrm{emri}}(t;T_{\mathrm{disk}}={10}^8\mathrm{yr})$ for different MBH masses $M_{•}$, where take ${\mu }_{\mathrm{cap}}=1$.

Figure 12

The EMRI rate $\mathrm{\Gamma }(t;T_{\mathrm{disk}}={10}^8\mathrm{yr})$ of the fiducial model is independent of the values of regularization parameters $\epsilon$ and $\eta$ we used in the main text.