Fully Relativistic Treatment of Extreme Mass-Ratio Inspirals in Collisionless Environments

TL;DR

The authors present a fully relativistic framework to model environmental effects on EMRIs in collisionless media, using a background Kerr spacetime and a distribution function $F(x,\\mathbf{p})=\\mu^{-3} f(\\mathcal{E},\\mathcal{C},\\mathcal{L}_z)$. At adiabatic order, they compute environment-induced rates $\\dot{\\varepsilon}_e$, $\\dot{l}_{z,e}$, and $\\dot{C}_e$ from the local four-acceleration $\\bm{a}$ and add these to GW-flux-driven evolution of geodesics described by $T_{\\rm orb}$ and $T_{\\rm rad}$ with $T_{\\rm orb} \sim M$ and $T_{\\rm rad} \sim M/q$. Applied to DM spikes around a Milky Way–like host with mass $M=10^6 M_{\\odot}$ and a representative $q=10^{-5}$, the fully relativistic waveforms show substantial environmental dephasing different from Newtonian extrapolations, with mismatches exceeding detector-threshold values for LISA after weeks to months of observation. The results demonstrate that fully relativistic treatment is essential for accurate EMRI waveform modeling in collisionless environments, with implications for LISA data analysis and new tests of general relativity; future work includes eccentric/inclined orbits, spinning secondaries, and backreaction on the environment.

Abstract

Future mHz gravitational wave (GW) interferometers will precisely probe massive black hole environments, such as accretion discs, cold dark matter overdensities, and clouds of ultralight bosons, as long as we can accurately model the dephasing they induce on the waveform of extreme mass-ratio inspirals (EMRIs). Most existing models rely on extrapolations from Newtonian results to model the interaction of the small black hole in an EMRI system with the environment surrounding the massive black hole. Here, we present a fully relativistic formalism to model such interaction with collisionless environments, focusing on the case of cold dark matter overdensities, like 'spikes' and 'mounds'. We implement our new formalism in the FastEMRIWaveforms framework and show that the resulting waveforms are significantly different from those based on a Newtonian treatment of environmental effects. Our results indicate that a fully relativistic treatment is essential to capture the environmental dephasing of GW signals from EMRIs accurately.

Figures (5)

• Figure 1: Energy loss per orbital cycle as a function of binary separation. Solid lines correspond to the fully relativistic model, the dashed to the Newtonian, and the dotted to including heuristic relativistic corrections as in Speeney:2022ryg, for an EMRI with mass-ratio $q=10^{-5}$ in a DM spike of a Milky-Way-like galaxy. An EMRI at $R = 12 M$ will merge after $\sim4$ years.
• Figure 2: Mismatch between relativistic and Newtonian environmental models as a function of observation time (in weeks before merger). Dashed lines mark the mismatch thresholds for various luminosity distances $D_L$, above which the Newtonian environmental waveform diverges significantly from the actual relativistic one, introducing systematics into LISA measurements. For an EMRI with mass-ratio $q=10^{-5}$ embedded in a dark-matter spike of a Milky-Way-like galaxy, the mismatch exceeds the critical threshold after only a few weeks of observation.
• Figure 3: Mismatch with respect to vacuum waveforms as a function of signal duration (in years before merger). The black and purple lines are for the benchmark $(q=10^{-5})$, or a slightly heavier companion.
• Figure 4: Distribution function in the companion frame at the ISCO ($R=6 M$) for the region $v^1\geq 0$ of a slice $v^2=0$. For a (quasi)circular EMRI, the distribution function is symmetric under $v^1\to-v^1$. The white region shows the loss cone of our fiducial DM spike; only the velocities in the colored regions correspond to stable bound orbits around the supermassive BH.
• Figure 5: Faithfulness between different models as a function of signal duration (in years before merger). The solid lines are for the benchmark $q=10^{-5}$ system, while the dashed are for a slightly heavier companion.