Dark matter distributions around extreme mass ratio inspirals: effects of radial pressure and relativistic treatment

Abstract: We investigate different treatments of dark matter (DM) distributions surrounding extreme mass ratio inspirals (EMRIs) to assess their impact on orbital evolution and gravitational wave emission. Density profiles derived from the mass current and from the energy-momentum tensor using a distribution function yield consistent results, but both differ substantially from profiles obtained using an anisotropic fluid model based on Einstein cluster ansatz. We find that the inclusion of radial pressure significantly modifies both the orbital dynamics and the resulting gravitational wave waveforms. By analyzing waveform dephasing and mismatches, we show that a fully relativistic treatment of DM distributions can substantially alter the detectability thresholds of DM halos. Our results demonstrate that radial pressure and relativistic modeling of DM are essential for accurately describing the dynamics and observational signatures of EMRIs embedded in DM halos.

• The paper shows that relativistic energy-momentum and mass-current profiles produce central dark matter densities up to six orders of magnitude larger than the Einstein-cluster model, while radial pressure reaches about 1% of the density near 10 black-hole masses.
• The paper finds that dynamical friction dominates halo-induced orbital evolution outside roughly 16 black-hole masses, accelerates inspiral, and exceeds corrections to the gravitational-wave flux across the modeled systems.
• The paper demonstrates that including relativistic profiles lowers the detectable halo compactness to about M/a₀ ≈ 10⁻⁵, while omitting radial pressure can create approximately one-radian dephasing and bias LISA parameter inference.

This paper examines how different relativistic treatments of dark matter (DM) distributions affect the orbital evolution and gravitational-wave (GW) signatures of extreme mass ratio inspirals (EMRIs) around massive black holes (MBHs). The authors construct three self-consistent prescriptions for a Hernquist-type DM halo surrounding a Schwarzschild MBH: a density derived from the mass current four-vector with vanishing radial pressure, a density and radial pressure extracted from the energy-momentum tensor built from the phase-space distribution function, and the widely used anisotropic-fluid profile based on the Einstein cluster ansatz. By evolving quasi-circular EMRIs under dynamical friction and GW radiation reaction in each background, they show that both radial pressure and fully relativistic halo modeling materially change waveform dephasing and detectability thresholds.

Relativistic dark matter profiles

The starting point is the Hernquist density $\rho_0(r) = Ma_0/[2\pi r(r+a_0)^3]$, from which the Eddington inversion yields the isotropic distribution function $f_{\rm HQ}(\epsilon_0)$. Following Sadeghian et al., the authors then obtain the general-relativistic density by adiabatically growing the MBH inside the halo: action variables are conserved during this growth, so the final distribution function is $f_{\rm fin}(\epsilon) = f_{\rm HQ}(E(\epsilon,L))$, with the energy remapped through the Schwarzschild radial action. Two densities follow from this construction:

• Mass-current density $\rho_{\rm J}$: obtained from $J^\mu = \rho_{\rm J} u^\mu_{\rm env}$, i.e., the time component of the mass current measured in the locally freely falling frame of the halo fluid.
• Energy-momentum density $\rho_{\rm T}$: obtained as $T_{\mu\nu}u^\mu u^\nu$ from the full second-moment integral over the distribution function, which also yields a nonzero radial pressure $p_r = T_{\mu\nu}k^\mu k^\nu$.

These are compared against the Einstein-cluster profile $\rho_{\rm C}$ of Cardoso et al., which assumes an anisotropic fluid with strictly zero radial pressure.

The central quantitative finding is that $\rho_{\rm T}$ and $f_{\rm HQ}(\epsilon_0)$0 agree closely — their ratio rises only to about 1.4 near the horizon — but both differ dramatically from $f_{\rm HQ}(\epsilon_0)$1. For compactness $f_{\rm HQ}(\epsilon_0)$2, the peak of $f_{\rm HQ}(\epsilon_0)$3 exceeds that of $f_{\rm HQ}(\epsilon_0)$4 by roughly six orders of magnitude; for $f_{\rm HQ}(\epsilon_0)$5, by roughly three orders. Near its peak, $f_{\rm HQ}(\epsilon_0)$6 scales as $f_{\rm HQ}(\epsilon_0)$7 and as $f_{\rm HQ}(\epsilon_0)$8, so it falls off faster than $f_{\rm HQ}(\epsilon_0)$9 at large radius while being far more centrally concentrated. The radial pressure is small but non-negligible: the ratio $f_{\rm fin}(\epsilon) = f_{\rm HQ}(E(\epsilon,L))$0 peaks near 0.01 at $f_{\rm fin}(\epsilon) = f_{\rm HQ}(E(\epsilon,L))$1 before dropping sharply toward the horizon. This result directly challenges the adequacy of the Einstein cluster ansatz for modeling the inner halo, since it underestimates the central density by many orders of magnitude in the parameter regime relevant to EMRIs.

Background geometry

For each prescription, the authors solve the static, spherically symmetric Einstein equations,

$f_{\rm fin}(\epsilon) = f_{\rm HQ}(E(\epsilon,L))$2

with boundary conditions matching the Schwarzschild horizon and asymptotic flatness (numerically imposed at $f_{\rm fin}(\epsilon) = f_{\rm HQ}(E(\epsilon,L))$3). The fiducial system is $f_{\rm fin}(\epsilon) = f_{\rm HQ}(E(\epsilon,L))$4, halo mass $f_{\rm fin}(\epsilon) = f_{\rm HQ}(E(\epsilon,L))$5, SCO mass $f_{\rm fin}(\epsilon) = f_{\rm HQ}(E(\epsilon,L))$6, with halo sizes $f_{\rm fin}(\epsilon) = f_{\rm HQ}(E(\epsilon,L))$7 to $f_{\rm fin}(\epsilon) = f_{\rm HQ}(E(\epsilon,L))$8.

Orbital evolution under dynamical friction and radiation reaction

The SCO's circular orbit evolves under two dissipative channels. Dynamical friction is modeled with the Chandrasekhar–Ostriker force including the relativistic correction factor $f_{\rm fin}(\epsilon) = f_{\rm HQ}(E(\epsilon,L))$9, which accounts for enhanced gravitational deflection and relativistic momenta of scattered DM particles; GW losses use the quadrupole formula evaluated on the curved background. Accretion onto the SCO is neglected.

Two structural results emerge from the flux comparison. First, GW radiation dominates the total energy loss for $\rho_{\rm J}$0, while dynamical friction dominates for $\rho_{\rm J}$1. Second, when the vacuum GW flux is subtracted to isolate the halo-induced correction $\rho_{\rm J}$2, the dynamical friction loss always exceeds it — the halo affects the inspiral primarily through drag rather than through modified GW emission. This ordering justifies treating dynamical friction as the leading environmental effect even in a relativistic treatment.

Numerically evolving $\rho_{\rm J}$3 shows that the DM halo accelerates the inspiral, with larger compactness producing faster decay. Among the three prescriptions, the orbital decay is fastest for the mass-current case ($\rho_{\rm J}$4, $\rho_{\rm J}$5), intermediate for the full energy-momentum case ($\rho_{\rm J}$6 with $\rho_{\rm J}$7), and slowest for the Einstein cluster case ($\rho_{\rm J}$8). The comparison between the first two isolates the effect of radial pressure: it slightly slows the inspiral relative to the pressureless relativistic model, a direct consequence of the $\rho_{\rm J}$9 term in $J^\mu = \rho_{\rm J} u^\mu_{\rm env}$0 modifying the metric and hence the binding energy and orbital frequency.

Waveforms and detectability

Waveforms are generated with the numerical kludge (NK) approach for circular orbits, using initial semi-latus rectum $J^\mu = \rho_{\rm J} u^\mu_{\rm env}$1, inclination $J^\mu = \rho_{\rm J} u^\mu_{\rm env}$2, luminosity distance $J^\mu = \rho_{\rm J} u^\mu_{\rm env}$3 Gpc, and one year of evolution. After one year, halo and vacuum waveforms are visibly distinguishable.

Detectability is assessed with two complementary diagnostics. The accumulated cycle difference $J^\mu = \rho_{\rm J} u^\mu_{\rm env}$4 (with a threshold of ~1 rad over one year) gives the following picture:

$J^\mu = \rho_{\rm J} u^\mu_{\rm env}$5	case 1 vs vac	case 2 vs vac	case 3 vs vac	case 1 vs 2	case 1 vs 3	case 2 vs 3
$J^\mu = \rho_{\rm J} u^\mu_{\rm env}$6	2604.0	2627.6	213.7	23.6	2390.4	2414.0
$J^\mu = \rho_{\rm J} u^\mu_{\rm env}$7	593.7	599.4	26.9	5.7	566.8	572.5
$J^\mu = \rho_{\rm J} u^\mu_{\rm env}$8	139.0	139.9	2.8	0.9	136.1	137.0
$J^\mu = \rho_{\rm J} u^\mu_{\rm env}$9	31.8	32.9	0.4	—	31.4	32.4

Under the Newtonian-based Einstein cluster profile (case 3), halos are detectable only for $\rho_{\rm T}$0; with either relativistic profile (cases 1 and 2), halos are detectable already for $\rho_{\rm T}$1. At $\rho_{\rm T}$2, the pairwise difference between cases 1 and 2 is itself ~1 rad, meaning that omitting radial pressure produces a dephasing at the detection threshold — so radial pressure must be included in accurate templates.

Mismatch calculations against the LISA noise spectral density reinforce this conclusion. With SNR ≈ 23.8 for the chosen source, the distinguishability threshold is $\rho_{\rm T}$3 for $\rho_{\rm T}$4 parameters:

$\rho_{\rm T}$5	case 1	case 2	case 3	case 1/2	case 1/3	case 2/3
$\rho_{\rm T}$6	0.984	0.979	0.957	0.810	0.948	0.928
$\rho_{\rm T}$7	0.931	0.951	0.883	0.425	0.860	0.915
$\rho_{\rm T}$8	0.843	0.841	0.638	0.328	0.832	0.887

All mismatches exceed the threshold down to $\rho_{\rm T}$9, including the case 1/case 2 mismatch (0.328 at $T_{\mu\nu}u^\mu u^\nu$0). The implication is twofold: relativistic treatment lowers the detectable halo compactness by roughly an order of magnitude relative to the Einstein cluster model, and even among relativistic models, neglecting radial pressure yields waveforms distinguishable at LISA sensitivity. Template families built on $T_{\mu\nu}u^\mu u^\nu$1 assumptions would therefore misattribute environmental effects or bias parameter inference.

Limitations and open questions

Several assumptions bound these conclusions. The analysis is restricted to quasi-circular orbits; eccentric EMRIs, which are astrophysically generic, may respond differently to the halo profile and radial pressure. The GW flux uses the quadrupole approximation on the curved background rather than black-hole perturbation theory or self-force calculations, and the waveforms rely on the NK scheme, so absolute mismatch values carry systematic uncertainty even though the relative comparisons among cases should be robust. Accretion onto the SCO is neglected, and the Coulomb logarithm is fixed at $T_{\mu\nu}u^\mu u^\nu$2. The halo is assumed to be stationary and spherically symmetric, with the distribution function invariant under adiabatic MBH growth — an assumption that breaks down if the halo is dynamically disturbed or if self-gravity of the spike becomes important. Finally, only a single fiducial system ($T_{\mu\nu}u^\mu u^\nu$3 + $T_{\mu\nu}u^\mu u^\nu$4, $T_{\mu\nu}u^\mu u^\nu$5) is studied, so the scaling of the detectability thresholds with source parameters remains to be established.

Conclusion

By solving the coupled Einstein–matter system for Hernquist-type halos with nonzero radial pressure and evolving EMRIs within the resulting backgrounds, this work establishes that the choice of relativistic treatment is not a subleading modeling detail. The mass-current and energy-momentum densities agree with each other but depart from the Einstein cluster profile by up to six orders of magnitude at the density peak; radial pressure, though bounded by $T_{\mu\nu}u^\mu u^\nu$6, shifts orbital decay rates enough to produce order-unity-radian dephasing; and relativistic modeling extends halo detectability down to compactness $T_{\mu\nu}u^\mu u^\nu$7, an order of magnitude below the threshold implied by the pressureless cluster model. These results indicate that waveform templates for EMRIs in dense environments should incorporate both the relativistic density profile and radial pressure to avoid systematic biases in future space-borne detector analyses.