Sharpening the dark matter signature in gravitational waveforms I: Accretion and eccentricity evolution

TL;DR

This paper addresses how dark matter (DM) spikes around massive black holes alter gravitational-wave (GW) signals from intermediate- and extreme-mass ratio inspirals. It advances the prior framework by incorporating DM particle accretion onto the orbiting companion and by extending the environmental feedback to general orbital eccentricities, validated against dedicated N-body simulations. The authors derive semi-analytical expressions for the accretion rate and accretion-induced backreaction, introduce a mass-conserving spike-depletion feedback, and develop orbit-evolution equations that couple $a$, $e$, and $m_2$ with GW, dynamical friction, and accretion. They demonstrate that accretion can produce substantial dephasing, often dominating at large separations, and that DM spikes tend to circularize eccentric orbits, with the combined effects shaping the GW phase in ways detectable by future space-based detectors. The work provides a robust framework for predicting DM-induced GW dephasing and paves the way for exploiting GW observations to probe DM spikes and their dynamics in galactic nuclei.

Abstract

Dark matter overdensities around black holes can alter the dynamical evolution of a companion object orbiting around it, and cause a dephasing of the gravitational waveform. Here, we present a refined calculation of the co-evolution of the binary and the dark matter distribution, taking into account the accretion of dark matter particles on the companion black hole, and generalizing previous quasi-circular calculations to the general case of eccentric orbits. These calculations are validated by dedicated N-body simulations. We show that accretion can lead to a large dephasing, and therefore cannot be neglected in general. We also demonstrate that dark matter spikes tend to circularize eccentric orbits faster than previously thought.

Figures (14)

• Figure 1: An intermediate mass ratio inspiral embedded within a dark matter spike. A black hole of mass $m_2$ orbits around a more massive black hole with mass $m_1 \gg m_2$, surrounded by dark matter. The orbit is characterised by the semi-major axis $a$ and orbital eccentricity $e$.
• Figure 2: Coordinate system for dark matter encounters with the companion. An illustration of a gravitational encounter with a particle in the secondary BH's rest frame. The initial incoming velocity of the DM particle is denoted as $\bm{V_0} = \mathbf{v} -\bm{u}$, and its impact parameter $b$ dictates its future; particles are absorbed within $b_\mathrm{acc}$ or their orbits perturbed up until $b_{max}$. The angle $\theta$ is the binary's true anomaly, while $\phi$ is between the separation $r_2$ and a vector pointing from the central black hole to the location of the impact parameter, $\bm{r} = \bm{r_2} +\bm{b}$.
• Figure 3: Coefficients for the accretion force $\mathcal{C}_\mathrm{acc}$ and mass accretion rate $\mathcal{C}_\mathrm{m}$ as a function of accretion radius $r_\mathrm{acc}$. The dimensionless accretion coefficients are defined in \ref{['eq:accretion_rates2']} and the surrounding text. Data points show the accretion coefficients estimated from $\dot{a}/a$ measured in simulations, while solid lines show the values calculated directly using \ref{['eq:coeffs_definition']}. In the top panel, the dashed green line shows the spurious contribution of dynamical friction to the accretion coefficient inferred from $\dot{a}/a$. For comparison, in the bottom panel, we also show the value of $\mathcal{C}_m'$ in \ref{['eq:spike_mass2']} as a dotted line.
• Figure 4: Orbit-averaged accretion coefficients $\left\langle \mathcal{C}_\mathrm{acc}(e) \right\rangle$ and $\left\langle \mathcal{C}_\mathrm{m}(e) \right\rangle$ as a function of orbital eccentricity $e$. Data points show the accretion coefficients estimated from $\dot{a}/a$ measured in simulations, while the solid lines show the values calculated by orbit-averaging of the coefficients defined in \ref{['eq:coeffs_definition']}. We fix the accretion radius to $r_\mathrm{acc} = 0.1\,a_i$.
• Figure 5: Feedback due to accretion as a function of accretion radius and eccentricity. In the upper panel, we show the ratio of the final to initial DM density profiles as a function of radius after 25 orbits. Solid lines show the results estimated from the NbodyIMRI simulations, while dashed lines are derived from the feedback model of \ref{['eq:df_accretion']} and \ref{['sec:massdepletion']}. Different colours correspond to different accretion radii $r_\mathrm{acc}$. In the lower panel, we fix the accretion radius to $r_\mathrm{acc} = 0.1\,a_i$ and show results for different eccentricities $e = \left\{0.0, 0.6, 0.8\right\}$. The coloured bars at the bottom of the panel show the range of radii traversed by the secondary BH over the orbits with eccentricities $e = 0.6$ and $e = 0.8$. The grey shading shows the region where the interactions between the DM particles and the central BH are softened.