Mass and spin coevolution of black holes inspiralling through dark matter

TL;DR

The paper develops a framework for mass and spin coevolution of a spinning black-hole companion inspiralling through a collisionless dark-matter spike. By deriving the DM accretion rate and angular-momentum transfer in Kerr geometry, it shows that spin-down and secular alignment with the orbital plane emerge, yielding a near-universal spin-evolution parameter $s \approx 2.8$ that is largely independent of local DM density and spike slope. Over astrophysical timescales, this coevolution leads to characteristic mass-spin correlations and spin-axis tilts that can be constrained by gravitational-wave observations, notably by LISA, offering a novel probe of dense DM environments. Observing rapidly spinning IMRI secondaries would disfavour dense DM spikes, providing complementary constraints to dynamical-friction-based inferences and enriching our understanding of DM near massive BHs.

Abstract

In extreme/intermediate-mass-ratio inspirals (E/IMRIs) embedded in dark-matter (DM) spikes, the secondary black hole can accrete collisionless particles from the surrounding halo. We study how the companion's spin controls this process, and the ensuing back-reaction on the magnitude and direction of the companion's spin vector. We find that higher spin suppresses the mass accretion rate but enhances the accretion-induced torques, driving spin-down and secular alignment of the companion's spin with the orbital plane. Collisionless DM accretion generically imprints a near-universal mass-spin correlation characterized by a spin-evolution parameter $s \simeq 2.8$, much larger than is the case for typical astrophysical environments, and largely independent of the local DM density and the spike slope. The associated spin-down proceeds on astrophysically relevant timescales, thus observations of rapidly spinning IMRI companions would disfavor the presence of dense DM environments, providing constraints complementary to those arising from dynamical friction.

Figures (6)

• Figure 1: Geometry of collisionless DM capture by a spinning companion BH. For a given relative velocity $\bm{V}$, the blue shaded region shows the set of impact parameters with $b\leq b_\mathrm{cr}(\chi,\tilde{a})$, that lead to capture. The capture contour is lopsided, with a larger effective radius for retrograde encounters, here corresponding to $\chi \in (-\pi/2, \pi/2)$. For illustration purposes, we choose $\bm V \parallel \hat{\bm x}$ and place the impact-parameter vector $\bm b$ in the $y$–$z$ plane.
• Figure 2: Accretion rate onto the companion black hole in an EMRI embedded in a dark matter spike, shown as a function of the companion’s dimensionless spin parameter. The blue points represent the numerical result of the quasi-Monte Carlo integration and associated 95% confidence intervals. The black dashed line depicts \ref{['eq:approx_results']}, while the dotted line shows the same expression but for a BH with $\tilde{a}=0$ (for comparison). The results are for an EMRI/IMRI at separation $r=10^3 R_M$ in a DM spike with $\gamma_\mathrm{sp} = 7/3$Gondolo_1999, and an angle between BH's velocity and spin-axis $\theta_u = \pi/3$.
• Figure 3: Rate of change of the companion’s spin as a function of its spin parameter for an EMRI in a dark matter spike. The colored dots denote the numerical results for the same configuration as in \ref{['fig:accretion_mass_force']}. The black dashed lines show the simpler approximate expression from \ref{['eq:dJdt_approx']}.
• Figure 4: Evolution of the spin–orbit tilt $\bm{\theta_L}$ as a function of the fractional mass growth $\bm{\Delta m/m_0}$. Solid lines show the full numerical integration of the coupled evolution of spin tilt and magnitude [from \ref{['eq:dthetaL_full', 'eq:dadt']}, together with \ref{['eq:approx_results']}]. Dashed lines correspond to the (leading order in $\tilde{a}$) analytic solution in \ref{['eq:approx_theta']}. At the top, we indicate the time to merge, assuming a quasi-circular inspiral driven by GW emission and accretion drag for the fiducial parameters \ref{['eq:fiducial']}.
• Figure 5: Evolution of the spin parameter $\bm{\tilde{a}}$ as a function of the fractional mass growth $\bm{\Delta m/m_0}$. On the right: solid lines represent $\tilde{a}(\Delta m)$ from numerically integrating \ref{['eq:s']}, dashed lines are the (leading order in $\tilde{a}$) solutions of \ref{['eq:s_bound']}, and dotted lines denote the $s=2$ solution. On the left: the fractional change in spin during the last few years before the merger. All results are for the fiducial parameters \ref{['eq:fiducial']} and a spin-orbit tilt $\theta_L = \pi/2$.