[Paper Review] Sharpening the dark matter signature in gravitational waveforms II: Numerical simulations with the NbodyIMRI code

Bradley J. Kavanagh, Theophanes K. Karydas+3 more|arXiv (Cornell University)|Feb 21, 2024

Geophysics and Gravity Measurements4 citations

TL;DR

This paper introduces NbodyIMRI, a publicly available N-body code that simulates intermediate and extreme mass ratio inspirals (IMRIs/EMRIs) in cold dark matter (DM) spikes, validating the standard dynamical friction formalism and quantifying the maximum impact parameter for effective scattering. It further identifies a 'stirring' effect from the time-varying binary potential that slows DM depletion, enhancing the DM-induced dephasing in gravitational waveforms and improving detection prospects for DM spikes in future observatories like LISA.

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.

Motivation & Objective

• To develop a high-accuracy N-body code for simulating binary black hole systems in cold dark matter spikes, enabling precise modeling of dynamical friction and feedback effects.
• To validate the standard dynamical friction formalism in non-uniform, finite-density DM environments typical of astrophysical spikes.
• To determine the maximum impact parameter $b_{\mathrm{max}}$ for effective scattering between DM particles and compact objects across varying mass ratios.
• To quantify the impact of a previously neglected 'stirring' effect from the time-dependent binary potential on DM distribution and depletion.
• To improve the accuracy of gravitational waveform modeling for future detection of dark matter signatures in LISA-like observatories.

Proposed method

• The NbodyIMRI code performs high-resolution N-body simulations of binary systems embedded in cold DM spikes, using large particle counts and adaptive timesteps to resolve dynamical processes.
• The code models the gravitational potential of the binary system using a multipolar expansion up to third order in $a/r$, capturing the time-varying potential due to orbital motion.
• The change in specific energy of DM particles due to the time-varying potential is computed via $\Delta\mathcal{E} = \int_0^{T_{\text{orb}}} \partial\psi/\partial t \, dt$, integrated along particle trajectories.
• A semi-analytical model is introduced to account for the 'stirring' effect, which redistributes energy in the DM spike without net loss, reducing depletion rates.
• The simulations are used to measure the effective dynamical friction force and assess feedback effects on the DM spike structure over time.
• The results are compared with analytical expectations to validate the standard dynamical friction formalism under realistic astrophysical conditions.

Experimental results

Research questions

• RQ1 Does the standard dynamical friction formalism hold in non-uniform, finite-density dark matter spikes around compact binaries?
• RQ2 What is the maximum impact parameter $b_{\mathrm{max}}$ for effective scattering between DM particles and a compact object across different mass ratios?
• RQ3 How does the time-varying potential of the binary induce a 'stirring' effect on the surrounding dark matter distribution?
• RQ4 To what extent does the stirring effect reduce dark matter depletion and thereby enhance the gravitational wave dephasing signature?
• RQ5 Can numerical simulations with high particle counts and fine timesteps reliably model feedback effects on the DM spike and improve waveform accuracy for future GW observations?

Key findings

• The standard dynamical friction formalism is validated in cold DM spikes, confirming its applicability despite the non-uniform and finite-density environment.
• The maximum impact parameter $b_{\mathrm{max}}$ for effective scattering is accurately determined across a range of mass ratios, providing a key input for waveform modeling.
• A 'stirring' effect is identified, where the time-varying binary potential redistributes energy in the DM spike without net loss, slowing down the depletion of DM particles.
• The stirring effect reduces the rate of DM depletion, thereby enhancing the gravitational wave dephasing signal and increasing the detectability of dark matter spikes.
• The NbodyIMRI code successfully captures both dynamical friction and feedback effects, enabling high-fidelity simulations crucial for future LISA-like gravitational wave observations.
• The net energy change in the DM spike due to stirring averages to zero over azimuthal angles, confirming that energy redistribution occurs without net extraction from the binary.