tokens: for Different Dark Matter Density Profiles in NSCs Figure 9 shows the final masses of IMBHs as a function of the initial ones. For those IMBHs that could deplete NSs during one Hubble time, the final masses include the contribution from all the stellar BHs and NSs in their host NSCs. Figure 9. Same as Figure 6 but for the IMBHs in NSCs. The circles, at about 1500 M ⊙ when α = 1.5 and 7800 when α = 2.0 and 7/3, mark the masses with which an IMBH could deplete all the stellar BH in the same cluster. The solid dots are the masses corresponding to NS depletion, which is about 1500 M ⊙ when α = 2.0 and 3200 M ⊙ when α = 7/3. 4.2. Merger Rate in Nuclear Star Clusters Per Mpc<sup>3</sup> The calculation of the merger rate per unit Mpc 3 for NSCs is almost the same as the previous calculation for GCs, but with two crucial modifications. (1) For the mass function of IMBHs, we consider two models. The first is suggested by Babak et al. ( 2017 ), which isAnother model, suggested by Gair et al. ( 2017 ), is more conservative,In this latter one, the exponent is positive, which significantly reduces the number of IMBHs. (2) When α ≥ 2.0, to calculate the M min in the integration of the mass function is determined by the solid dots in Figure 9 , because smaller IMBHs in NSCs would have depleted all the NSs. The corresponding merger rates for BHs and NSs per Mpc 3 are presented in Table 6 . Again, we find an enhancement of BH and NS merger rates in the presence of DM minispikes. Table 6 The Merger Rate per Mpc 3 of IMRIs or EMRIs with BHs R BH and Neutron Stars R NS Formed by IMBH for Different Dark Matter Density Profiles From the calculations we can also find that the results are sensitive to the BH and NS supply. With DM minispikes, the duty cycle of BHs and NSs may be shorter than the Hubble time, as the dramatically increased merger rate can lead to a high efficiency of consumption of compact objects. As a result, the appearance nowadays strongly depends on the total BH and NS supply. 5. Could Stellar Dynamics Deplete the Minispikes? We have shown in Figures 1 – 3 that dynamical hardening, by ejecting intruding stars, is important for the initial evolution of IMRI/EMRIs, when pc. It is worth discussing here whether the same process could eject DM particles and hence deplete the DM minispikes. The difference between hardening against DM and hardening against stars is that the interloper stars that we considered are gravitationally unbound to the IMBHs but the DM particles are deep in the potential well of the IMBH. The DM particles are more difficult to deplete. To show this more clearly, we calculate the hardening timescale associated with DM using the ejection timescale derived in Sesana et al. ( 2008 ) for the stars gravitationally bound to binary MBHs. They showed that it is , where P 0 is the orbital period of the binary. For example, the pink solid line in Figure 10 illustrates the dependence of this timescale on the IMBH mass. Figure 10. Same as
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