Black holes surrounded by generic matter distributions: Polar perturbations and energy flux

We develop a numerical approach to compute polar parity perturbations within fully relativistic models of black hole systems embedded in generic, spherically symmetric, anisotropic fluids. We apply this framework to study gravitational wave generation and propagation from extreme mass-ratio inspirals in the presence of several astrophysically relevant dark matter models, namely the Hernquist, Navarro-Frenk-White, and Einasto profiles. We also study dark matter spike profiles obtained from a fully relativistic calculation of the adiabatic growth of a BH within the Hernquist profile, and provide a closed-form analytic fit of these profiles. Our analysis completes prior numerical work in the axial sector, yielding a fully numerical pipeline to study black hole environmental effects. We study the dependence of the fluxes on the DM halo mass and compactness. We find that, unlike the axial case, polar fluxes are not adequately described by simple gravitational-redshift effects, thus offering an exciting avenue for the study of black hole environments with gravitational waves.

Figure 1

Mass profiles for some representative DM configurations considered in this work. The NFW model is truncated at the cutoff radius $r_c=5a_0$. The Hernquist, NFW and Einasto profiles are obtained by the “cutoff factor” recipe $\rho (r)\to (1-4M_{\mathrm{BH}}/r)\rho (r)$. The “DM spike” profile is obtained with the fully relativistic calculation of the adiabatic growth of a BH within a Hernquist profile from Refs. [12, 38], as described in the main text. Going from the top to the bottom panels, the DM halo mass grows relative to the central BH mass ($M_{\text{Halo}}/M_{\mathrm{BH}}=1$, 10, 100), while the compactness of the halo decreases going from the left to the right panels ($M_{\text{Halo}}/a_0=0.1$, 0.01, 0.001). The vertical lines correspond to a representative orbital radius $r_p=7.9456M_{\mathrm{BH}}$ that will be used in Table 1 below to compare GW energy fluxes.

Figure 2

Solid lines: $\ell =m=2$ mode GW energy fluxes, in units of $q^2=m_p^2/M_{\mathrm{BH}}^2$, in the presence of a DM spike matter background. The frequency range on the $x$ axis corresponds to particle orbital radii ranging from $r_p=50M_{\mathrm{BH}}$ to $r_p=6M_{\mathrm{BH}}$. In the left panel we compare DM spike halos with the same value of $M_{\text{Halo}}/M_{\mathrm{BH}}=100$ but different compactness $M_{\text{Halo}}/a_0$, while in the right panel we compare DM spike halos with fixed compactness $M_{\text{Halo}}/a_0=0.01$ but different values of $M_{\text{Halo}}/M_{\mathrm{BH}}$. For reference, the dashed line shows the $\ell =m=2$ multipolar component of the vacuum energy flux.

Figure 3

Solid lines: relative difference between the $\ell =m=2\mathrm{GW}$ energy fluxes, in units of $m_p^2/M_{\text{Halo}}^2$, in vacuum (${\dot{E}}_v$) and in the presence of a matter background (${\dot{E}}_m$) for different total halo masses, fixing the compactness to $M_{\text{Halo}}/a_0=1/100$. This frequency range corresponds to particle orbital radii ranging from $r_p=50M_{\mathrm{BH}}$ to $r_p=6M_{\mathrm{BH}}$. Dashed lines; relative difference between ${\dot{E}}_m$ and the vacuum fluxes redshifted according to Eq. (19). Dot-dashed lines; relative difference between ${\dot{E}}_m$ and the post-Newtonian fluxes of Eq. (20).

Figure 4

Solid lines: relative difference between the $\ell =m=2\mathrm{GW}$ energy fluxes in vacuum (${\dot{E}}_v$) and in the presence of matter (${\dot{E}}_m$) for different compactness, fixing the halo mass to $M_{\text{Halo}}=100M_{\mathrm{BH}}$. This frequency range corresponds to particle orbital radii ranging from $r_p=50M_{\mathrm{BH}}$ to $r_p=6M_{\mathrm{BH}}$. Dashed lines; relative difference between ${\dot{E}}_m$ and the vacuum fluxes redshifted according to Eq. (19). Dot-dashed lines; relative difference between ${\dot{E}}_m$ and the post-Newtonian fluxes of Eq. (20).

Figure 5

Multipolar components of the GW energy flux in the presence of a DM spike background as a function of the particle’s orbital radius $r_p/M_{\mathrm{BH}}$. We fix the halo mass to $M_{\text{Halo}}/M_{\mathrm{BH}}=100$ and the compactness to $M_{\text{Halo}}/a_0=0.001$. Solid (dashed) lines refer to polar (axial) modes.