Post-adiabatic waveforms from extreme mass ratio inspirals in the presence of dark matter

Abstract

Extreme mass-ratio inspirals (EMRIs), in which a solar mass compact object is whirling around a supermassive black hole, act as precise tracers of the spacetime geometry and astrophysical environment around the supermassive black hole. These systems are highly sensitive to even the smallest deviations from the vacuum general relativity scenario. However, detecting these signals requires highly accurate waveform modeling up to the first post-adiabatic order, incorporating self-force effects, system parameters, and environmental influences. In this paper, we focus on the impact of dark matter on gravitational waveforms. Cold dark matter in galactic centers can be redistributed by the gravitational pull of a supermassive black hole, forming a dense, spike-like profile. When an EMRI evolves in such an environment, the interaction between the binary and the surrounding dark matter can leave distinctive imprints on the emitted waveform, and thus offer a novel way to probe the nature and distribution of dark matter. We specifically examine how dark matter modifies the background spacetime. By treating these modifications perturbatively, we present a framework to incorporate dark matter environmental effects into gravitational waveform modeling at the first post-adiabatic order.

Figures (11)

• Figure 1: Dark matter spike density profile (blue line) for the cases where the initial profile (red line) is Hernquist (left) and NFW (right). In this analysis, we assume a total dark matter mass of $M_{\rm Halo} = 10^4 M_{\rm BH}$, with $M_{\rm Halo}/a_0 = 10^{-3}$, and a cutoff radius given by $r_{\rm c} = 100 M_{\rm Halo} a_0 / M_{\rm BH}$.
• Figure 2: The plots of the mass function $m(r)$ (left panel) and the redshift function $q(r)$ (right panel) as functions of radius are shown for the Hernquist (solid red line) and NFW (blue dashed line) dark matter density profiles, using the fitting parameters listed in \ref{['tab:Fitting_Parameters']}. Since the dark matter density vanishes for $r \leq 4M_{\textrm{BH}}$, the mass function $m(r)=M_{\textrm{BH}}$ in this region. Beyond $4M_{\textrm{BH}}$, $m(r)$ gradually increases and asymptotically approaches to $M_{\textrm{BH}}+M_{\textrm{Halo}}$ near the cutoff radius. The redshift function $q(r)$ remains small for both the Hernquist and NFW profiles and approaches a constant value near the cutoff radius.
• Figure 3: The plot of radial shift in fixed frequency formalism as a function of orbital frequency. Here, the red line represents the relativistic result whereas the dashed line represents its Newtonian counterpart. For low orbital frequency values, there is a significant radial shift. At higher frequencies, however, the enclosed dark matter mass becomes negligible, resulting in a reduced radial shift.
• Figure 4: The plots for the rescaled axial $\tilde{\Psi}_R^{(1,0)}$ (for $l=2,~m=1$ mode) and polar $\tilde{\Psi}_Z^{(1,0)}$ (for $l=2,~m=2$ mode) master functions as functions of the compactified radial coordinate. The particle is assumed to be located at $\sigma = 0.2$ (corresponding to $r_\Omega = 10M_{\textrm{BH}}$), indicated by the black dashed line. Due to the distributional nature of the source term, both master functions exhibit a discontinuity at the particle’s location.
• Figure 5: The numerical convergence of the axial master function $\Psi_R^{(1,0)}$ is shown as a function of numerical resolution $N$. A high-resolution reference solution $\Psi_R^{(1,0),\textrm{ref}}$ computed with $N^{\textrm{ref}}=150$ is used to assess convergence. The left panel displays the convergence behavior for various values of $r_\Omega$ at fixed angular mode $l=2,~m=1$, while the right panel shows the convergence at fixed $r_\Omega = 10M_{\textrm{BH}}$ across different angular modes. In both cases, we observe rapid numerical convergence. However, the convergence rate slows for larger $r_\Omega$ and higher $l$ modes. This decline is due to the steepening gradient of the master function in these regimes, requiring finer resolution (i.e., more grid points) to maintain accuracy.