Extreme-mass-ratio inspirals in relativistic accretion discs

TL;DR

This paper investigates how relativistic Lindblad torques affect extreme-mass-ratio inspirals (EMRIs) embedded in relativistic accretion discs around Kerr black holes. It extends Hirata's relativistic Lindblad torque framework to include disc structure, derives mode-by-mode torques from Teukolsky-based amplitudes, and introduces a pressure regularization to obtain convergent total torques. The results reveal that relativistic corrections amplify torques by orders of magnitude, alter the radial scaling with a spin-dependent slope $n_r$, and can even reverse the torque near the ISCO with the reversal radius shifting inward for higher spins; these signatures provide a potential discriminant of disc configurations with LISA. The work suggests LISA could directly probe the midplane structure of inner accretion discs by measuring the relativistic Lindblad torque signatures in golden EMRI events, thereby linking GW observations to disc physics in the strong-field regime.

Abstract

We compute relativistic Lindblad torques for circular, equatorial extreme-mass-ratio inspirals (EMRIs) embedded in relativistic thin accretion discs, including spinning black hole configurations. We find that relativistic effects can amplify the magnitude of these torques by orders of magnitude in the strong-field regime, and that the torque can even reverse direction as the EMRI approaches the innermost stable circular orbit (ISCO). However, we show that the location of this reversal is highly spin-dependent, shifting progressively closer to the ISCO, where gravitational-wave emission completely dominates the inspiral, as the spin of the central black hole increases. Spin also modifies the radial dependence of the Lindblad torques. We investigate whether Lindblad torques can be approximated by parametrised power laws of the form T_LR = A(r_s / 10M)^n_r (or combinations thereof), and find significant spin- and disc-dependent variations in the slope parameter n_r. For instance, for spin a/M = 0.9, we find n_r = 3.6 in the strong-field regime, compared to the Newtonian value of n_r = 4.5. Given current forecasts of parameter recovery for ``golden'', loud EMRIs in accretion discs (Δn_r ~ 0.5), we predict LISA could distinguish between different disc configurations through their relativistic Lindblad torque signatures, providing the first direct probe of the midplane structure of the inner region of accretion discs, which is inaccessible to electromagnetic observations.

Figures (6)

• Figure 1: Ratio of the relativistic Lindblad torques to the gravitational-wave torques for an equatorial, circular EMRI embedded in different disc configurations, shown as a function of orbital separation $r_s$ and for representative disc models with the same viscosity and accretion rate (see Fig. \ref{['fig:discProfile']}). For spinning primaries, we use the relativistic Novikov-Thorne disc model. Dotted segments indicate regions where the Lindblad torque becomes negative.
• Figure 2: Radial location of Lindblad resonances, $r_\text{LR}$, as a function of the secondary orbital radius, $r_s$, for different azimuthal mode numbers $m$ and primary BH spin $a$. In the large-$m$ limit, the inner and outer Lindblad resonances become equidistant and close to the secondary. For large orbital radii, outer Lindblad resonances are consistently located closer to the secondary than their inner counterparts (for the same $m$), but this hierarchy reverses at smaller $r_s$. The critical radius at which this switch occurs is spin-dependent, decreasing with increasing BH spin parameter $a$.
• Figure 3: Normalised Lindblad torques as a function of the EMRI radius for different modes $m$. For large $r_s\gg r_+$, $\mathcal{N}^{(m)}$ approaches its constant Newtonian limit. For smaller $r_s$, the inner resonances become stronger than the outer ones, as anticipated from their location in Fig. \ref{['fig:LRlocations']}. For large $m$, the amplitudes of the inner and outer torques become comparable, as the resonances tend to become equidistant from the secondary. Increasing primary spin leads to smaller normalised Lindblad torques.
• Figure 4: Left panel: Variation of the mode-by-mode inner and outer normalised Lindblad torques with $m$. The growth becomes quadratic for any secondary radius $r_s$ and spin $a$ at relatively small $m$. Right panel: same as in the left panel but for the total torque, i.e., the sum of the inner and outer contributions. In this case, the growth is approximately linear with $m$ because the quadratic components of the outer and inner torques cancel each other.
• Figure 5: Surface density (left) and aspect ratio (right) radial profiles for the Novikov--Thorne disc model at different BH spins. The physical parameters of the system are $M = 10^6 \, M_\odot$ for the primary mass, $\dot{M}/\dot{M}_\text{Edd}= 0.1$ for the accretion rate in terms of the Eddington limit, and $\alpha = 0.1$ for the phenomenological parameter controlling viscosity. Changing these values alters the overall normalisation of the profiles but not their radial dependence. Dashed blue curves show the corresponding Shakura--Sunyaev power law profiles. Profiles are plotted to large radius, beyond the inner disc region---where radiation pressure dominates--- illustrating that relativistic effects are noticeable already for $r \lesssim 250M$.