EMRI Dephasing from a Torsion-Inspired Near-Zone Kerr Deformation: Motivated by Spin-Polarized Dark Matter

Abstract: Extreme-mass-ratio inspirals (EMRIs) are sensitive probes of weak conservative perturbations in the strong-field region of massive black holes. We study a phenomenological EMRI model motivated by Einstein--Cartan gravity in which a spin-polarized dark-matter spike is described by a Weyssenhoff fluid. After torsion is eliminated algebraically, the local spin contribution contains a repulsive exterior source $$U_{tt}^{\rm spin}\propto-σ<em>0<sup>2/r<sup>3$$. Solving the corresponding static linearized field equation, however, does not produce a global $1/r^3$ metric perturbation; the response contains a mass renormalization, a logarithmic $r<sup>{-1}$ tail, and an $M/r<sup>2$ term. We therefore introduce $$g</em>{μν}^{\rm eff}=g_{μν}^{\rm Kerr}+αh_{μν}^{\rm eff}$$ only as a local near-zone matching ansatz, not as a complete rotating Einstein--Cartan black-hole solution. Within this torsion-inspired deformation we compute circular equatorial inspirals and analytic-kludge waveforms. The fiducial model can produce large phase shifts in an idealized adiabatic calculation, but the forecast is optimistic and does not include a full LISA/Taiji response, Teukolsky/self-force fluxes, eccentricity, inclination, or high-dimensional parameter degeneracies. The results should be read as constraints on an effective near-zone operator rather than as a prediction of minimally coupled Einstein--Cartan dark matter.

• The paper demonstrates that torsion-induced corrections to the Kerr metric can produce significant EMRI phase shifts, with dephasing reaching nearly 966 rad over 10^4–10^5 cycles.
• Analytic-kludge models reveal an outward ISCO shift of approximately 3.7×10⁻³ M and an earlier plunge by about 5000 M under specific dark matter-induced spin conditions.
• The study highlights that LISA/Taiji-class detectors may effectively probe dark-matter microphysics through accumulated gravitational waveform dephasing in EMRI systems.

EMRI Dephasing from Torsion-Induced Kerr Near-Zone Deformations: Spin-Polarized Dark Matter Motivation

Background and Physical Motivation

The paper investigates secular gravitational-wave dephasing in extreme-mass-ratio inspirals (EMRIs) due to a torsion-induced near-zone deformation of the Kerr metric, with a phenomenological motivation from spin-polarized dark-matter spikes around supermassive black holes. Traditional EMRI analyses treat the compact secondary as a test-body inspiraling in a Kerr geometry, with adiabatic evolution governed by gravitational radiation reaction. However, the paper focuses on conservative perturbations by invoking Einstein-Cartan gravity, wherein intrinsic spin density from a polarized dark-matter spike sources spacetime torsion, ultimately producing a local repulsive spin-spin interaction.

The macroscopic modeling utilizes the Weyssenhoff spin fluid to represent a dark matter spike with polarization amplitude following $o(r) \propto r^{-3/2}$, paralleling steep strong-field density profiles. The algebraic elimination of torsion yields an effective $U_{\text{spin}} \propto -o^2/r^3$ repulsive source, distinct from the purely attractive mass-density perturbations considered in prior environmental EMRI studies.

Effective Metric Deformation and Analytical Framework

The deformation is introduced via an effective near-zone ansatz for the metric:

$$g_{\mu\nu}^{\text{eff}} = g_{\mu\nu}^{\text{Kerr}} + a\, h_{\mu\nu}^{\text{eff}}$$

with the dominant term in $h_{\mu\nu}^{\text{eff}}$ scaling as $r^{-3}$, matching local algebraic torsion effects. Crucially, the full metric solution to the static field equation does not globally carry an $r^{-3}$ mode but yields a mass renormalization, logarithmic tail, and $M/r^2$ contribution. The adopted ansatz isolates the short-range repulsive force relevant in the strong-field regime near the ISCO, rather than attempting a full Einstein-Cartan rotating black-hole solution.

Circular equatorial orbits and analytic-kludge waveforms are computed, with the deformation parameter $a$ phenomenologically matched to the microscopic spin density and normalized according to the expected dark-sector amplitude.

Orbital Dynamics: ISCO Shift, Plunge Advancement, and Dephasing

The spin-torsion-induced metric perturbation yields an outward shift of the ISCO, representing the principal conservative effect. For optimized parameter settings (e.g., $M=10^6 M_\odot$, $a=0.9$, $U_{\text{spin}} \propto -o^2/r^3$0, $U_{\text{spin}} \propto -o^2/r^3$1), the ISCO is displaced by $U_{\text{spin}} \propto -o^2/r^3$2 relative to the general relativity baseline. Although the orbital correction is perturbatively small, it leads to an earlier plunge by $U_{\text{spin}} \propto -o^2/r^3$3, corresponding to several hours in physical time.

The critical observable is not the instantaneous orbital variation, but the accumulated gravitational-wave phase over $U_{\text{spin}} \propto -o^2/r^3$4--$U_{\text{spin}} \propto -o^2/r^3$5 cycles. In the fiducial scenario, the phase difference reaches $U_{\text{spin}} \propto -o^2/r^3$6 rad, significantly exceeding the one-radian threshold for waveform resolvability. This demonstrates that weak, localized spin-torsion operators can induce substantial phase-coherent EMRI signatures under favorable density and polarization assumptions.

Detector Response, Parameter Estimation, and Degeneracy Structure

Adiabatic analytic-kludge waveforms are compared to the predicted sensitivity of LISA/Taiji-class detectors. The characteristic strain for the torsion-inspired EMRI falls within the mHz band, aligning with the optimal detector window. Noise-weighted overlaps and waveform mismatches are computed in the $U_{\text{spin}} \propto -o^2/r^3$7 parameter space, with mismatches exceeding $U_{\text{spin}} \propto -o^2/r^3$8 for benchmark parameter values, indicating effective distinguishability under idealized conditions.

The Fisher information matrix analysis reveals substantial but non-exact degeneracy between the spin-torsion strength $U_{\text{spin}} \propto -o^2/r^3$9 and Kerr spin $$g_{\mu\nu}^{\text{eff}} = g_{\mu\nu}^{\text{Kerr}} + a\, h_{\mu\nu}^{\text{eff}}$$0. Both parameters alter the ISCO and late-inspiral frequency evolution; however, their radial scaling produces distinct phase patterns due to integration over the full inspiral domain. Observable constraints are ultimately placed on the combination $$g_{\mu\nu}^{\text{eff}} = g_{\mu\nu}^{\text{Kerr}} + a\, h_{\mu\nu}^{\text{eff}}$$1, highlighting the quadratic dependence on dark-matter spike density.

Comparisons with Environmental and Modeling Systematics

The impact of the torsion-inspired deformation is compared with other EMRI environmental and modeling systematics:

• Ordinary dark-matter spikes: Attractive and degenerate with mass-density corrections.
• Dynamical friction and accretion: Dissipative, also producing secular phase drifts.
• Gas disk migration: Hydrodynamic torques yield both stochastic and secular dephasing.
• Conservative short-range deformations: Different radial scaling but strong degeneracy.
• Self-force and Teukolsky fluxes: GR systematics dominate precision phase corrections.
• Instrumental response and parameter fitting: Degrades sensitivity and broadens posteriors.

Waveform mismatches and parameter separation may be compromised by these effects, indicating that robust identification requires full modeling of EMRI environmental channels and higher-order perturbative corrections.

Practical and Theoretical Implications

On the practical side, the results provide a strong motivation for incorporating and constraining effective spin-torsion operators in EMRI waveform analyses targeting LISA/Taiji data. The phase-coherent nature of EMRIs renders them highly responsive to minute strong-field perturbations, including those arising from dark-sector microphysics.

Theoretically, the framework establishes a concrete link between microscopic spin structure in dark matter and macroscopic gravitational observables. The algebraic nature of torsion sources in Einstein-Cartan gravity and the quadratic coupling in the Weyssenhoff fluid together open novel channels for probing dark-matter properties in the vicinity of supermassive black holes.

Further studies should address:

• Extension to generic EMRIs with eccentric and inclined orbits.
• Incorporation of full Teukolsky/self-force waveforms and high-order radiative fluxes.
• Integration of more detailed dark-matter models with calculable polarization profiles.
• Bayesian parameter estimation with realistic detector response, environmental effects, and high-dimensional parameter spaces.

EMRI phase drift due to torsion-inspired deformations is a theoretically motivated diagnostic for near-horizon structure and dark-sector spin effects—provided the phenomenological parameters can be solidly anchored to underlying microphysics.

Conclusion

This paper rigorously formulates and analyzes a torsion-inspired near-zone Kerr deformation motivated by spin-polarized dark matter, providing analytic and numerical evidence that such a mechanism can produce substantial accumulated dephasing in LISA/Taiji-band EMRIs under optimistic assumptions. The results serve as an effective operator-level constraint, not as a unique prediction for minimally coupled Einstein-Cartan dark matter. Realistic observational tests will require more advanced waveform models, environmental systematics, and connection to microscopic dark-sector physics, but the current analysis identifies the relevant scaling, phenomenological matching, and waveform systematics for future studies.