The Depletion of Collisionless Dark Matter Spikes

Abstract

Dense concentrations of dark matter (DM) surrounding black holes provide a compelling opportunity to probe the nature of DM. In the classic Gondolo-Silk model, the adiabatic growth of a massive black hole (MBH) in a DM cusp produces a steep density spike ($ρ\propto r^{-7/3}$), potentially inducing measurable gravitational-wave dephasings in intermediate and extreme mass-ratio inspirals (IMRIs/EMRIs). We challenge this paradigm by considering a collisionless spike embedded in a realistic nuclear star cluster (NSC). Using 1D orbit-averaged Fokker-Planck (FP) simulations of isotropic NSCs, we show that mass segregation in a multi-mass stellar cusp accelerates relaxation, relative to single-mass models, thereby driving the DM to the lower density $r^{-3/2}$ Bahcall-Wolf profile within $\lesssim 1 \mathrm{Gyr}$. In the inner regions, where the FP description breaks down, we model strong triple interactions between DM particles and EMRIs using post-Newtonian 3-body simulations. We show that EMRIs eject DM particles via slingshots, depleting the inner spike over a few Gyrs. Because EMRI number densities are too low to drive two-body relaxation, and collisionless DM cannot efficiently repopulate the depleted phase space, this depletion is irreversible. While the extent of EMRI-induced depletion depends on the EMRI rate and mass, we find reductions in DM densities by several orders of magnitude. Hence, DM-induced dephasings for EMRIs may fall below the detectability threshold of LISA for MBHs at $z = 3$ (2.14 Gyr) with masses $\lesssim 10^{5}\,M_\odot$ (for an $\mathcal{O}(10) \, \mathrm{Gyr}^{-1}$ EMRI rate), extending to $\lesssim 10^6\,M_\odot$ for more optimistic rates of $\mathcal{O}(300-1000) \, \mathrm{Gyr}^{-1}$. Our findings substantially reduce the parameter space over which MBHs can host detectable collisionless DM spikes.

Figures (12)

• Figure 1: Schematic illustration of our system. The DM spike is shown in brown and the stellar sphere in blue. The full DM distribution (not shown) extends into the outer boundary of the stellar sphere and far beyond it. In reality, the loss cone is not a sphere but a cone in semi-major axis--eccentricity space.
• Figure 2: Time evolution of the DM and stars, whose mass function follows our extended model, around an MBH of mass $4.3 \times 10^6 M_{\odot}$, integrated over 2 Gyr. From left to right, the panels show: the DM two-body relaxation time-scale, the DM density profile as a function of radius, the DM density power-law slope $\gamma_{\rm DM} = -\mathrm{d}\ln \rho_{\rm DM}/\mathrm{d}\ln r$, and the density power-law slope of the most massive stellar species. The different colours represent evolution time in units of Gyr as indicated in the legend in the bottom right panel. The top and bottom rows display our multi-mass model and single-effective-mass model, respectively. The vertical black dot–dashed line marks the MBH radius of influence, the horizontal dashed line in the left panels denotes one Hubble time (14 Gyr), and the horizontal dot–dashed lines in the right side panels denote the relaxed BW profiles. Clearly, the multi-mass model exhibits significantly faster relaxation compared to the single-mass case.
• Figure 3: Example three-body evolution showing a DM particle being ejected via a gravitational slingshot interaction with an EMRI.
• Figure 4: The EMRI initial periapsis distribution (Eq. \ref{['eq:f_EMRI']}) per logarithmic interval for $M_{\rm BH} = [10^4, 10^5, 10^6, 2 \times 10^6, 10^7]\,M_{\odot}$ at $t = 2.14$ Gyr. The left cut-off is determined either by the maximum semi-major axis, taken to be the RoI of the MBH, or $f_{\mathfrak{r}_{\rm p, 0}} (\mathfrak{r}_{\rm p, 0} < 2R_{\rm BH}) = 0$. The right cut-off arises as there does not exist any $0 \leq e_{\rm LC}< 1$ such that $T_{\rm GW} = T_J$, i.e. the GW-induced inspiral is too slow compared to phase space diffusion.
• Figure 5: Ejection fraction of DM particles as a function of inclination for different mass ratios $q$ (top panel), and the same curves rescaled by $q^2$ (bottom panel). The collapse of the rescaled curves demonstrates the expected $q^2$ scaling (see Appendix \ref{['appsec:Scaling with Mass Ratio']}). In all cases we fix $\mathcal{E} = 0.2$, $\mathfrak{R}_{\rm p} = 10R_{\rm BH}$, $\mathfrak{e} = 0.4$, $\mathfrak{r}_{\rm p} = 12R_{\rm BH}$, and $M_{\rm BH}=4.3 \times 10^6 M_{\odot}$ (these ejection fractions are independent of $M_{\rm BH}=4.3 \times 10^6 M_{\odot}$, as discussed) while randomising the DM particle's orbital orientation and phase.