Probing Quantum Gravity effects with Extreme Mass Ratio Inspirals around Rotating Hayward Black Holes

We investigate extreme mass-ratio inspirals (EMRIs) around a rotating Hayward black hole to assess the detectability of signatures arising from quantum gravity.The quantum parameter $α_0$, which encodes deviations from general relativity (GR), introduces extra correction terms in both the orbital frequency and the fluxes. Our results show that after one year of accumulated observation, these corrections induce a detectable dephasing in the EMRI waveform. Using the modified orbital evolution driven by $α_0$, we generate waveforms via the augmented analytic kludge (AAK) model implemented in the \texttt{FastEMRIWaveforms} package. Furthermore, we utilize the time-delay interferometry (TDI) to suppress the laser noise and phase fluctuations induced by spacecraft motion, and then employ the Fisher information matrix (FIM) to test the sensitivity of LISA in detecting deviations from GR. Our results demonstrate the potential of LISA to probe quantum-gravity effects through high-precision observations of EMRIs.

💡 Research Summary

This paper investigates the prospect of detecting quantum‑gravity signatures using extreme‑mass‑ratio inspirals (EMRIs) that orbit a rotating Hayward regular black hole (HBH). The Hayward metric replaces the central singularity with a Planck‑scale nonsingular core, characterized by a dimensionless quantum‑deviation parameter α₀; when α₀=0 the spacetime reduces to the Kerr solution. By applying the Newman‑Janis algorithm, the authors obtain a rotating HBH metric that depends on both the spin a and the quantum parameter α₀, revealing a novel coupling between rotation and quantum corrections that is absent in the static case.

The study restricts the small compact object to equatorial motion (θ=π/2), allowing the Carter constant to vanish. Conserved energy E and axial angular momentum L_z are expressed in terms of the metric functions, and the orbital dynamics are parametrized by the semi‑latus rectum p and eccentricity e via the usual relation r = p/(1+e cos χ). Analytic expressions for E(p,e) and L_z(p,e) are derived, reducing to the Kerr formulas when the mass function m(r) equals unity. Fundamental frequencies Ω_r and Ω_φ are split into a general‑relativistic (GR) part and a quantum‑corrected (QC) contribution proportional to α₀: Ω_i = Ω_i^GR + α₀ Ω_i^QC (i = r, φ). The QC terms depend non‑trivially on a, e, and p, and become larger for higher spin, indicating that rotation amplifies the sensitivity of the orbit to quantum effects.

Radiation reaction is treated in the adiabatic approximation. The total energy and angular‑momentum fluxes are written as ⟨dE/dt⟩ = ⟨dE/dt⟩_GR + α₀⟨dE/dt⟩_QC and similarly for L_z. The GR fluxes are taken from high‑order post‑Newtonian results supplemented by Teukolsky‑based fits. For the QC part the authors employ the quadrupole‑octupole formalism, constructing mass quadrupole I_ij, current quadrupole J_ij, and mass octupole M_ijk from the particle trajectory, and then deriving explicit α₀‑dependent expressions (Eqs. 16‑17) that contain powers of (1–e²) and inverse powers of p, with additional spin‑dependent terms proportional to a.

Using the adiabatic equations d p/dt and d e/dt obtained from the balance law (Eqs. 21), the authors integrate the orbital evolution over a one‑year observation window. They define a dephasing measure ΔΨ_i = 2 T_obs ΔΩ_i and focus on the azimuthal component ΔΦ ≈ ΔΨ_φ, because the radial contribution is negligible. Assuming a detection threshold of ΔΦ ≈ 1 rad for a signal‑to‑noise ratio of ~30 (typical for LISA), they compute ΔΦ as a function of α₀ for several spin values. Figure 2 shows that for a=0.2, α₀ as low as ~10⁻³ yields a dephasing above the threshold, while lower spins require larger α₀. This demonstrates that the combined effect of rotation and quantum corrections can produce observable phase shifts within LISA’s sensitivity.

To translate these orbital effects into observable waveforms, the authors employ the Augmented Analytic Kludge (AAK) model implemented in the FastEMRIWaveforms (FEW) package. With fixed source parameters {a=0.1, p₀=11, e₀=0.1, M=10⁶ M_⊙, μ=10 M_⊙}, they generate plus‑polarization h₊ waveforms for several α₀ values. Figure 3 illustrates that after one year the α₀‑induced modifications lead to clear deviations from the Kerr‑based waveform, confirming that EMRI signals can encode quantum‑gravity information.

Instrumental noise is addressed by constructing first‑generation Time‑Delay Interferometry (TDI) observables, which cancel laser frequency noise arising from unequal arm lengths and spacecraft motion. The authors then build a Fisher information matrix (FIM) for the parameter set {α₀, a, p₀, e₀, …} and evaluate the expected statistical errors. The resulting 1‑σ uncertainty on α₀ is found to be ≲10⁻³ for the chosen configuration, well below the dephasing detection threshold, indicating that LISA could place meaningful constraints on the quantum‑deviation parameter.

In conclusion, the paper provides a comprehensive pipeline—from a theoretically motivated regular black‑hole spacetime, through modified geodesic dynamics and radiation‑reaction fluxes, to realistic waveform generation, TDI processing, and Fisher‑matrix based parameter estimation—demonstrating that rotating Hayward black holes offer a viable laboratory for probing quantum‑gravity effects with future space‑based gravitational‑wave detectors. The authors suggest extensions such as incorporating inclined orbits, higher‑order spin couplings, and multi‑parameter Bayesian analyses to further refine the prospects of testing quantum gravity with EMRIs.