Bayesian inference for tidal heating with extreme mass ratio inspirals

TL;DR

The paper tackles testing near-horizon dissipation in EMRIs by constraining the horizon reflectivity $|\mathcal{R}|^2$ using a fully Bayesian analysis on equatorial eccentric EMRIs with relativistic 0PA waveforms and the LISA response. It models tidal heating through a Kerr-like ECO framework where reflectivity modifies horizon flux via $(1-|\mathcal{R}|^2)$ and solves the Teukolsky equations to produce accurate waveforms. Injection-recovery studies at a two-year, $\rho=50$ SNR show that strong-field configurations yield tight bounds on $|\mathcal{R}|^2$ at the $10^{-3}$–$10^{-4}$ level and that neglecting tidal heating biases intrinsic parameters, validating horizon-dissipation tests with EMRIs. The results demonstrate the necessity of a full 13D Bayesian treatment and relativistic templates over PN approximations, establishing EMRIs as precision probes of black-hole horizon physics for future space-based GW missions.

Abstract

Extreme mass ratio inspirals (EMRIs) provide unique probes of near-horizon dissipation through the tidal heating. We present a full Bayesian analysis of tidal heating in equatorial eccentric EMRIs by performing injection-recovery studies and inferring posterior constraints on the reflectivity parameter $|\mathcal{R}|^2$ while sampling in the full EMRI parameter space. We find that in the strong-field regime the posterior uncertainties are smaller, indicating a stronger constraining capability on the tidal heating. Using two-year signals with an optimal signal-to-noise ratio (SNR) of $ρ=50$, EMRIs can put bounds on $|\mathcal{R}|^2$ at the level of $10^{-3}$--$ 10^{-4}$ for a rapidly spinning central object. Moreover, we show that neglecting the tidal heating can induce clear systematic biases in the intrinsic parameters of the EMRI system. These results establish EMRIs as promising precision probes for detecting and constraining black hole event horizons.

Figures (6)

• Figure 1: The change to the inspiral trajectory arising from the tidal heating. We fix the evolution time to $T=2~\mathrm{yr}$ and compare two runs with the reflectivity $|\mathcal{R}|^2=0$ and $|\mathcal{R}|^2=0.01$ over the initial-condition grid $(p_i,e_i)$ for $(M,\mu,a)=(10^6 M_\odot,\,10\,M_\odot,0.95M)$. The four panels are arranged as follows: upper left and upper right show the fractional change in the final semi-latus rectum $10^4\times(p_f(0.01)-p_f(0))/p_f(0)$ and the final eccentricity $10^4\times(e_f(0.01)-e_f(0))/e_f(0)$, and lower left and lower right show the relative change in the accumulated radial phase $10^4\times(\Phi_r(0.01)-\Phi_r(0))/\Phi_r(0)$ and the azimuthal phase $10^4\times(\Phi_\phi(0.01)-\Phi_\phi(0))/\Phi_\phi(0)$, respectively.
• Figure 2: The comparison of the 5PN waveform (the reflectivity $|\mathcal{R}|^2=0$) with the adiabatic waveform ($|\mathcal{R}|^2=0$ and $|\mathcal{R}|^2=1$). Here $(M,\mu,a,p,e) = (10^{6}M_\odot, 10M_\odot, 0.95M, 13M, 0.4)$. The sky and viewing angles are $(q_k,\phi_k,q_s,\phi_s) = (\pi/4, \pi/3, \pi/2, \pi/3)$, the luminosity distance is $D_L = 3$ Gpc, and the initial orbital phases are $(\Phi_{\phi0},\Phi_{r0}) = (\pi/3, \pi)$. The left panel shows the initial GW waveform, and the right panel shows the waveform after $2\times10^{6} \,\mathrm{s}$.
• Figure 3: The posterior distributions for six parameters obtained from the MCMC analysis in the full 13-dimensional parameter space (blue) and in a reduced 6-dimensional parameter space (red) with the optimal ${\rm SNR}=50$. The injected values are listed in Table \ref{['tab:priors']}. For the 13-dimensional run, we sample the full 13-dimensional parameter space, but only the six parameters of interest are shown and the remaining seven parameters are omitted for clarity. For the 6-dimensional run, the remaining seven parameters are fixed to their injected values, and the sampling is performed only in the 6-dimensional subspace. The posteriors are inferred from a simulated two-year LISA observation. The vertical and horizontal lines mark the injected true value of each parameter, and the contours denote the 68.3%, 95.4%, and 99.7% confidence intervals.
• Figure 4: The posterior distributions for the EMRI source parameters obtained from the MCMC analysis with the injected value of the reflectivity $|\mathcal{R}|^{2}=0$ and optimal ${\rm SNR}=50$. The parameters varied in the inference are the MBH spin $a$, initial semi-latus rectum $p_i$ and initial eccentricity $e_i$, while the injected values of other parameters and the priors are listed in Table \ref{['tab:priors']}. The posteriors are inferred from a simulated two-year LISA observation. The vertical and horizontal lines mark the injected true value of each parameter, and the contours denote the 68.3%, 95.4%, and 99.7% confidence intervals.
• Figure 5: The best $68.3\%$ upper bound on the reflectivity $|{\cal R}|^2$ as a function of initial eccentricity $e_{i}$ with the initial semi-latus rectum $p_{i}$, i.e., $8.4M$ (blue), $8.6M$ (orange), $8.8M$ (green) and $9.0M$ (red), respectively. The injected values of all other parameters are listed in Table \ref{['tab:priors']}.