Sharpening the dark matter signature in gravitational waveforms II: Numerical simulations with the NbodyIMRI code

TL;DR

The paper addresses how dark-matter spikes around black holes influence gravitational-wave signals from intermediate- and extreme-mass-ratio inspirals by modelling dynamical friction with high-precision N-body simulations. It introduces NbodyIMRI, a publicly available code that represents the DM spike with pseudo-particles, computes forces from the binary BHs with softening, and advances the system with a fixed-timestep leapfrog integrator while ignoring PN terms to focus on short-timescale dynamics. The authors calibrate the dynamical-friction force, determine the maximum effective impact parameter $b_{\max}$ to be about $0.31$ times the binary separation, and reveal a stirring effect from the time-dependent binary potential that redistributes DM and sustains friction. They show that including stirring improves agreement with simulations and that HaloFeedback captures only the order-of-magnitude of spike depletion, underscoring the need to include time-dependent effects for accurate GW waveform modelling and DM spike physics.

Abstract

Future gravitational wave observatories can probe dark matter by detecting the dephasing in the waveform of binary black hole mergers induced by dark matter overdensities. Such a detection hinges on the accurate modelling of the dynamical friction, induced by dark matter on the secondary compact object in intermediate and extreme mass ratio inspirals. In this paper, we introduce NbodyIMRI, a new publicly available code designed for simulating binary systems within cold dark matter `spikes'. Leveraging higher particle counts and finer timesteps, we validate the applicability of the standard dynamical friction formalism and provide an accurate determination of the maximum impact parameter of particles which can effectively scatter with a compact object, across various mass ratios. We also show that in addition to feedback due to dynamical friction, the dark matter also evolves through a `stirring' effect driven by the time-dependent potential of the binary. We introduce a simple semi-analytical scheme to account for this effect and demonstrate that including stirring tends to slow the rate of dark matter depletion and therefore enhances the impact of dark matter on the dynamics of the binary.

Figures (10)

• Figure 1: Schematic of the force calculations in NbodyIMRI. The gravitational force on body $B$ due to body $A$ is indicated by an arrow ($A \rightarrow B$). The code includes unsoftened force calculations between the two BHs, $m_1$ and $m_2$. The force on the DM pseudo-particles $\tilde{m}_\mathrm{DM}$ due to the two BHs is softened, as described in the text. No pair-wise forces are calculated between DM particles, as the gravitational potential due to the DM spike is expected to be subdominant compared to that of the binary.
• Figure 2: Velocity distribution of DM particles in the spike. The blue curve shows the velocity distribution in the rest frame of the central black hole $m_1$, while the orange curve shows the velocity distribution relative to the orbiting compact object $m_2\ll m_1$. The results shown here are at a distance $r = 100 \,R_\mathrm{ISCO}$ from the central BH of mass $m_1 = 1000\,M_\odot$. We assume the orbiting object is on a circular orbit with velocity $v_\mathrm{circ}(r) = \sqrt{G m_1/r}$. In the rest frame of the central BH, the escape velocity of DM particles is $v_\mathrm{max}(r) = \sqrt{2G m_1/r}$. The dashed blue line is the velocity distribution when the density profile is not truncated ($r_t \rightarrow \infty$).
• Figure 3: Stability of a simulated DM spike (in the absence of a secondary BH). We simulate a central BH with mass $m_1 = 1000\,M_\odot$ surrounded by a DM spike. The interactions with the central BH are softened with $\epsilon_1 = 10\,R_\mathrm{ISCO}$ (grey shaded region). The simulation is run for the equivalent of 1000 orbits, for a binary with separation $a = 100\,\mathrm{isco}$.
• Figure 4: Decay of semi-major axis due to DM dynamical friction in $N$-body simulations. Each of the 16 pale green curves shows the fractional change in the semi-major axis in a single realisation, simulating a circular binary with masses $(m_1, m_2) = (1000, 1)\,M_\odot$ in a DM spike, with a DM softening length of $\epsilon_2 = 10^{-3} a_i$. The thick curve shows the average over the ensemble of realisations. The decay in the semi-major axis is driven by dynamical friction with the DM spike, and the rate of decay allows us to infer the strength of the dynamical friction force, characterised by $\mathcal{C}_\mathrm{DF}$.
• Figure 5: Dynamical friction coefficient $\mathcal{C}_\mathrm{DF}$ estimated from $N$-body simulations. Each point in the upper panels shows the DF coefficient estimated from the orbital decay $\Delta a/a$ of an ensemble of simulated binaries, assuming a specific value of the softening length of the secondary object $\epsilon_2$. We follow each binary for 25 orbits, simulating the DM spike with $N_\mathrm{DM} = 256\,\mathrm{k}$ particles. The point highlighted with a circle in the top-centre panel corresponds to the set of simulations shown in \ref{['fig:SemimajorAxis']}. Each of the stars in the lower panel shows the value of $b_\mathrm{max}$ obtained from a fit to the DF coefficients in the panel above for the corresponding mass. The dashed black lines in the upper panels show the values of $\mathcal{C}_\mathrm{DF}$ calculated from \ref{['eq:DF_coefficient']} for these best fitting $b_\mathrm{max}$ values (with errors shown as grey shaded regions). The horizontal dashed line in the lower panel shows the mean value across different masses: $b_\mathrm{max} = 0.3 \,r_2$.