Mass-gap extreme mass ratio inspirals

In this work, we propose a new subclass of extreme-mass-ratio-inspirals (EMRIs): mass-gap EMRIs, consisting of a compact object in the lower mass gap $\sim (2.5-5)M_{\odot }$ and a massive black hole (MBH). The mass-gap object (MGO) may be a primordial black hole or produced from a delayed supernova explosion. We calculate the formation rate of mass-gap EMRIs in both the (dry) loss-cone channel and the (wet) active galactic nucleus disk channel by solving Fokker-Planck-type equations for the phase-space distribution. In the dry channel, the mass-gap EMRI rate is strongly suppressed compared to the EMRI rate of stellar-mass black holes (SBHs) as a result of mass segregation effect. In the wet channel, the suppression is roughly equal to the mass ratio of SBHs over MGOs, because the migration speed of a compact object in an active galactic nucleus disk is proportional to its mass. We find that the wet channel is much more promising to produce mass-gap EMRIs observable by spaceborne gravitation wave detectors. (Non)detection of mass-gap EMRIs may be used to distinguish different supernova explosion mechanisms and constrain the abundance of primordial black holes around MBHs.

Figure 1

In the fiducial stellar cluster around a MBH with $M_{•}=4\times {10}^6{M}_{\odot }$, distribution functions $f_i(E,R)$ ($i=\mathrm{star},\mathrm{mgo},\mathrm{sbh}$) at $t_{\mathrm{f}}=2\mathrm{Gyr}$ are shown in the first 3 panels. The 4th panel shows $R$-integrated distribution functions ${\overline{f}}_i$ at $t_{\mathrm{i}}=0$ (dashed lines) and at $t_{\mathrm{f}}=2\mathrm{Gyr}$ (solid lines). All the distribution functions are shown in units of ${10}^5{\mathrm{pc}}^{-3}/(2\pi {\sigma }^2{)}^{3/2}$.

Figure 2

Upper panel: EMRI rates ${\mathrm{\Gamma }}_{\mathrm{emri},\mathrm{lc}}$ and prompt infall rates ${\mathrm{\Gamma }}_{\mathrm{infl},\mathrm{lc}}$ of both SBHs and mass-gap objects in the loss-cone channel. Lower panel: differential rates $d\mathrm{\Gamma }/dE$ at $t=2\mathrm{Gyr}$, and the two dots are the critical energy $E_{\mathrm{gw}}^{\mathrm{mgo},\mathrm{sbh}}$.

Figure 3

Average EMRI rates ${\overline{\mathrm{\Gamma }}}_{\mathrm{emri},\mathrm{dry}}$ and average prompt infall rates ${\overline{\mathrm{\Gamma }}}_{\mathrm{infl},\mathrm{dry}}$ of both SBHs and mass-gap objects in the loss-cone channel.

Figure 4

The migration timescales of a $10M_{\odot }$ BH embedded in the fiducial $\alpha$ and $\beta$ disks, respectively, where the GW emission becomes dominant at $E/{\sigma }^2\sim {10}^5$ or equivalently $a\sim {10}^2{M}_{•}$.

Figure 5

In the fiducial stellar cluster around a MBH with $M_{•}=4\times {10}^6{M}_{\odot }$, the cluster-component distribution functions $f_i(E,\mu )$ ($i=\mathrm{star},\mathrm{mgo},\mathrm{sbh}$) at $t={10}^7\mathrm{yr}$ are shown in the first 3 panels. The 4th panel shows $\mu$-integrated cluster-component distribution functions ${\overline{f}}_i(E)$ (solid lines) and disk-component $g_i(E)$ (dashed lines). All the distribution functions are shown in units of ${10}^5{\mathrm{pc}}^{-3}/(2\pi {\sigma }^2{)}^{3/2}$.

Figure 6

Wet EMRI rates ${\mathrm{\Gamma }}_{\mathrm{emri},\mathrm{disk}}(t)$ of SBHs and MGOs. For comparision, we also plot the average dry EMRI rates ${\overline{\mathrm{\Gamma }}}_{\mathrm{emri},\mathrm{lc}}$ during the quiet phase as horizontal lines (Fig. 3).

Figure 7

The surface number density $\mathrm{\Sigma }(r)$ of disk-component SBHs at different times for the two different AGN disk models.