Gravitational waves of extreme-mass-ratio inspirals in a rotating black hole with Dehnen dark matter halo

Abstract: Extreme Mass Ratio Inspirals (EMRIs) are among the key targe sources for the space-based gravitational wave (GW) detectors. The waveforms of the EMRIs are highly sensitive to the types of the central supermassive black hole (SBH) and can serve as a novel sensitive tool to probe the background spacetime. In this work, we compute GWs radiated from EMRIs in the backgrounds of Kerr black hole and rotating black hole with Dehnen-type dark matter halo (DMBH). Following the Teukolsky prescription, we obtain the perturbed equations for curvature tensor from the Newman-Penrose (NP) equations, and for the DMBH we obtain the radial and angular equations through separation of variables. To solve the equations with numerical method we apply the Sasaki-Nakamura (SN) transformation to convert the Teykolsky-type equation into the SN equation. We study the radiation reaction of GWs by computing the energy flux and angular momentum flux at infinity and at the horizon. The orbital evolution is then derived from the total fluxes. We extract the two polarizations of GWs by solving the equation numerically. By comparing the waveforms of Kerr and DMBH, it is found that the DM halo induces noticeable changes in both the amplitude and phase of GWs. We compute the strain of GW detector with the response function and evaluate the mismatch between the waveforms of Kerr and DMBH. The results show that the mismatch increases with the mass parameter of DM halo and the spin of the SBH.

• The paper develops a Teukolsky–Sasaki–Nakamura framework for computing adiabatic gravitational waveforms, fluxes, and LISA responses from inclined circular EMRIs in rotating black holes surrounded by Dehnen dark matter halos.
• The paper finds that halo-induced amplitude and phase changes produce distinguishable waveforms when the dimensionless halo mass parameter exceeds approximately 1.3×10⁻⁴ for a signal-to-noise ratio of 20, while mismatch is largely insensitive to whether the halo is cored or cuspy.
• The paper shows that increasing black-hole spin generally strengthens waveform differences, suggesting rapidly rotating supermassive black holes and LISA EMRIs could provide sensitive probes of dark matter near galactic centers, subject to adiabatic and circular-orbit approximations.

Overview

This paper computes gravitational waveforms from extreme-mass-ratio inspirals (EMRIs) around a rotating black hole embedded in a Dehnen-type dark matter halo (DMBH), and quantifies how distinguishable such waveforms are from their Kerr counterparts for a LISA-like detector. The central object is a Kerr-like metric obtained via the Newman–Janis prescription, where the mass function $\mathcal{M}(r)$ acquires a halo contribution controlled by three parameters: the halo mass parameter $\tilde{k}$, the scale radius $r_c$, and the profile index $\tilde{\sigma}\in[0,2)$, with $\tilde{\sigma}=0$ corresponding to a cored profile and $\tilde{\sigma}=1$ to a cuspy one. The authors work entirely within black hole perturbation theory, following the Teukolsky formalism, and produce time-domain waveforms, fluxes, orbital evolution, detector strains, and waveform mismatches.

Perturbation framework

Because metric perturbations of rotating backgrounds are difficult to formulate directly, the authors construct decoupled equations for the Newman–Penrose curvature scalars $\psi_0$ and $\psi_4$ using the NP formalism. Starting from three coupled NP equations involving $\psi_0$, $\psi_1$, $\tilde{k}$0 and the spin coefficients $\tilde{k}$1, $\tilde{k}$2, they exploit the type-D identities $\tilde{k}$3 and $\tilde{k}$4 to eliminate $\tilde{k}$5 and obtain master equations of the form $\tilde{k}$6, with the $\tilde{k}$7 equation obtained by tetrad interchange. Substituting the DMBH spin coefficients yields an explicit Teukolsky-type equation whose structure parallels the Kerr case but with $\tilde{k}$8 replaced by the radial function $\tilde{k}$9 and its derivatives throughout.

Separation of variables with $r_c$0 produces an angular equation identical to the Kerr spin-weighted spheroidal harmonic equation — so the angular sector is unaffected by the halo — while the radial equation carries all halo dependence through $r_c$1. The asymptotic solutions at infinity diverge because the potential is long-ranged, obstructing direct numerical integration. The authors therefore apply the Sasaki–Nakamura (SN) transformation, deriving explicit short-range potentials $r_c$2 and $r_c$3 and the transformed coefficients $r_c$4. A notable structural result is that in the large-$r_c$5 expansion of the SN solution, the coefficients $r_c$6 and $r_c$7 coincide exactly with the Kerr values, while only $r_c$8 acquires halo-dependent terms proportional to $r_c$9; the SN solution reduces to Kerr as $\tilde{\sigma}\in[0,2)$0.

Orbital dynamics and radiation reaction

The secondary's motion is treated with the Hamilton–Jacobi formulation, which remains integrable since the DMBH spacetime is Petrov type D. For inclined circular orbits, the polar motion is regularized by the substitution $\tilde{\sigma}\in[0,2)$1, removing turning-point divergences. The polar period $\tilde{\sigma}\in[0,2)$2 is expressed analytically through complete elliptic integrals, and the azimuthal advance per polar period $\tilde{\sigma}\in[0,2)$3 is obtained in closed form involving elliptic integrals of the first, second, and third kinds. The harmonic frequencies are then $\tilde{\sigma}\in[0,2)$4, which discretize the Green's-function amplitudes $\tilde{\sigma}\in[0,2)$5 and $\tilde{\sigma}\in[0,2)$6.

Energy and angular momentum fluxes at infinity follow from the Isaacson stress-energy tensor applied to the large-$\tilde{\sigma}\in[0,2)$7 behavior of $\tilde{\sigma}\in[0,2)$8. Horizon fluxes are computed via the Hawking–Hartle method: the horizon area increase induced by the shear $\tilde{\sigma}\in[0,2)$9 of null generators gives the energy flux per solid angle, with the DMBH-specific factor $\tilde{\sigma}=0$0 accounting for the modified first law. The relation between horizon and infinity amplitudes is fixed by a generalized Teukolsky–Starobinsky identity, whose coefficients now depend on $\tilde{\sigma}=0$1. Summing over modes yields total fluxes, and imposing the circular-orbit conditions $\tilde{\sigma}=0$2, $\tilde{\sigma}=0$3 produces evolution equations $\tilde{\sigma}=0$4 and $\tilde{\sigma}=0$5 driven by the total fluxes, closing the adiabatic inspiral scheme.

Numerically, the authors solve the SN equation using Hughes' algorithm, determining both the effective numerical horizon radius and infinity via Richardson-type extrapolation iterated until successive differences fall below $\tilde{\sigma}=0$6, rather than truncating at a fixed large radius. Waveforms are reconstructed as

$\tilde{\sigma}=0$7

with spheroidal harmonics evaluated using the Black Hole Perturbation Toolkit, truncated at $\tilde{\sigma}=0$8. Representative parameters are $\tilde{\sigma}=0$9, $\tilde{\sigma}=1$0, spin $\tilde{\sigma}=1$1, and inclination $\tilde{\sigma}=1$2. The comparison shows that the halo induces both amplitude changes and phase shifts relative to Kerr.

Detector response and mismatch analysis

The LISA strain is computed as $\tilde{\sigma}=1$3 with the standard long-wavelength response functions $\tilde{\sigma}=1$4, $\tilde{\sigma}=1$5, including the full time dependence of the source direction and polarization angles due to LISA's annual orbital motion. Waveform distinguishability is quantified by the mismatch $\tilde{\sigma}=1$6, where the overlap uses the noise-weighted inner product with the Robson–Cornish–Liu LISA noise power spectral density. Adopting the standard indistinguishability criterion $\tilde{\sigma}=1$7 with signal-to-noise ratio $\tilde{\sigma}=1$8 and $\tilde{\sigma}=1$9 parameters for the DMBH model, two waveforms are considered distinguishable when $\psi_0$0.

The main quantitative findings are:

Effect	Result
Halo mass parameter $\psi_0$1	Mismatch increases monotonically; waveforms become distinguishable for $\psi_0$2 (at $\psi_0$3)
Profile shape ($\psi_0$4)	Mismatch essentially insensitive to cored vs. cuspy profile; same threshold $\psi_0$5 for both
Spin $\psi_0$6	GW amplitude decreases and phase shift grows with spin; mismatch increases overall with spin (non-monotonically over extended range)

Two implications follow directly. First, the detectability threshold on $\psi_0$7 provides a concrete target: EMRI observations could constrain or detect halos above roughly $\psi_0$8 in the dimensionless mass parameter, independent of whether the underlying profile is cored or cuspy. Second, the stronger distinguishability at higher spins implies that EMRIs onto rapidly spinning SBHs are more effective probes of dark matter environments.

Limitations and open questions

Several assumptions bound these results. The treatment is adiabatic: radiation reaction enters only through leading-order fluxes, neglecting conservative self-force corrections and second-order dissipative effects that are known to matter for precision EMRI templates. Only inclined circular orbits are considered; eccentric and generic orbits, which dominate realistic EMRI populations, are not modeled. The halo is assumed static and spherically distributed in the equatorial-symmetric form inherited from the Newman–Janis construction, ignoring dynamical friction, accretion-driven halo evolution, and possible back-reaction of the secondary on the halo (e.g., wake formation). The mismatch analysis fixes most system parameters and explores only $\psi_0$9 and $\psi_4$0; a full parameter-estimation study with degeneracies between halo parameters and intrinsic binary parameters remains open. Finally, the insensitivity of the mismatch to $\psi_4$1 raises the specific question of whether any EMRI observable can discriminate cored from cuspy profiles, or whether profile shape is fundamentally degenerate with other parameters at this order.

Conclusion

This paper extends the Teukolsky-based EMRI waveform machinery to rotating black holes surrounded by Dehnen dark matter halos, providing the full pipeline from perturbation equations through SN transformation, geodesic orbits, adiabatic fluxes, and LISA response to waveform mismatches. The key results are that the halo leaves measurable amplitude and phase imprints, that distinguishability sets in above $\psi_4$2 independently of profile shape, and that higher central spins enhance sensitivity to the halo. Within its stated approximations — adiabatic circular-inclined orbits and a static halo — the work establishes a quantitative basis for using space-based EMRI observations to probe dark matter distributions near supermassive black holes.