Extreme mass ratio inspirals from the Hills mechanism, The eccentricity distribution of Hills EMRIs in LISA

Author(s)

Sersante, Biancamaria, Mancieri, Davide, Zavala, Kevin Leopoldo Muñoz, Broggi, Luca, Penoyre, Zephyr, Rossi, Elena Maria, Sesana, Alberto

Abstract

Extreme mass ratio inspirals (EMRIs) are promising sources for future space-based gravitational-wave detectors. We investigate EMRIs produced through the Hills mechanism, in which a stellar-mass black-hole binary (BBH) is tidally separated by a massive black hole (MBH), leaving one component bound to it. We present the first self-consistent modelling of this channel, combining an orbit-averaged Fokker-Planck treatment of the nuclear star cluster (NSC), three-body modelling of the binary-MBH encounter, Monte Carlo evolution of captured black holes (BHs), and fully relativistic EMRI trajectories to follow the evolution from binary supply down to final plunge. We find that Hills EMRIs are substantially more circular than classical EMRIs formed through two-body relaxation. For an MBH of $4 \times 10^6 M_\odot$ and circular, equal-mass BBHs of $20 M_\odot$ with initial separation of $0.1$ AU, EMRI mean eccentricities are approximately $0.097$ at our adopted LISA-band entry threshold and $0.034$ at plunge, compared with $0.478$ and $0.228$, respectively, for classical EMRIs. Of the initial BBHs, approximately $0.4$ per cent become EMRIs, whereas approximately $60$ per cent of captured BHs do. For comparable NSC initial compact-object mass, we find approximately $1.6 \times 10^{-8}$ Hills EMRIs per year per galaxy, approximately $12$ per cent of the classical EMRI yield. The Hills channel contribution to the overall eccentricity distribution depends strongly on the binary mass fraction in NSCs. We discuss implications for identifying EMRI formation channels with LISA and for quasi-periodic eruptions.

Figures

Caption

Cartoon to illustrate the various steps of our analysis of Hills EMRIs. At the end of step 1, binaries can either remain to populate the NSC or they can cross the tidal radius (the fraction of BBHs crossing $r_{\rm t}$ is $f^{r_{\rm t}}_{\rm BBH}=38,550/3,720,000=1.04$ per cent). Binaries that cross the tidal radius can either separate (in $64$ per cent of the cases) or survive ($\approx 36$ per cent of our systems). Binaries that separate result in a component being bound to the MBH and the other ejected. The bound component can be an EMRI, an SE or a DP. The total fraction of EMRIs produced by the initial population is $f^{\rm H}_{\rm EMRIs}=0.39$ per cent (see Section \ref{results}). The total fraction of captured BHs resulting in an EMRI is $59$ per cent. Systems that are EMRIs evolve, driven by GW-emission, until plunge. Their final properties are the result of Step 4.

Caption

Scatter distributions of binaries that cross $r_{\rm t}$ (38550 systems) coloured by the number of BBH per logarithmic unit of semi-major axis and $(1-e)$, $n_{\rm BBH}$, (\emph{plasma} colormap), with $25,40,50,75-$ per cent contours (black solid lines). Scatter distribution of captured BHs (24592 systems), coloured by the number of BHs, per logarithmic unit of semi-major axis and $(1-e)$, captured via Hills mechanism ($n_{\rm cap}$) (\emph{Spectral} colormap). The corresponding $25,40,50,75-$ per cent contours (purple) are lines of constant PDF. The black diagonal line represents the loss-cone boundary ($r_{\rm BH}=8R_g$) and orbital elements are here computed in the Newtonian framework. $r_{\rm BH}$ is the threshold in Newtonian phase space below which BHs plunge into the MBH. The corresponding relativistic definition of the loss cone boundary can be found in Sec. \ref{step3section} The pink diagonal line marks $t_{\rm GW}=t_{\rm rlx}$ (at early times), computed as Eq. (\ref{tgw}) and Eq. (\ref{trlx}), respectively. The horizontal dashed line marks the characteristic semi-major axis for a separating binary $\alpha_D$, while the vertical dotted lines mark the characteristic change in eccentricities for separating binary, $\delta_D=1-e_{\rm D}$ for $\beta=1,10,100$ from right to left (as we are starting approximately on a parabolic trajectory, $\delta_D$ coincides with the final characteristic eccentricity value reached by the bound component upon binary separation).

Caption

Orbital properties of the binaries reaching the tidal disruption radius, in the potential of the MBH only. \emph{Leftmost plot,first panel}: CDF of the diving factor $\beta$. \emph{Leftmost plot, second panel}: CDF of the eccentricity of the binary's CM, $e_{\rm cm}$, along its trajectory around the MBH. The solid dark line (light dashed) is the CDF for binaries on orbits bound (unbound) to the MBH. The dashed vertical line marks $10^{-3}$ where the CDFs of bound and unbound orbits are, respectively, 0.98 and 1. \emph{Leftmost plot, third panel}: CDF of the loss-cone diffusivity parameter $q$. The vertical dashed-dotted line corresponds to $q=1$, indicating the approximate boundary between the full and empty loss cone regime. In all panels, the gray horizontal dotted line marks the $50$ per cent probability. \emph{Rightmost plot}: Scatter plot of the CM-eccentricity vs $\beta$, coloured by $q$. The black solid line corresponds to $q=1$. We left out $0.27$ per cent of the systems, with $\beta>100$, for visualization purposes in the leftmost and rightmost panels.

Caption

Scatter plot of the orbital properties (Newtonian semi-major axis and eccentricity) of the orbits of the Hills-captured BHs. The points are coloured according to the corresponding initial $\beta$ (\emph{left panel}), to the cosine of initial inclination (\emph{central panel}) and by $q$ (\emph{right panel}). The pink solid line marks $t_{\rm GW}=t_{\rm rlx}$ (at early times) and the black diagonal line marks the boundary of the MBH loss cone, $r_{\rm BH}=8R_{\rm g}$. Horizontal and vertical lines are the same as in Fig. (\ref{fig:total})

Caption

Examples of orbit evolution in the $(1-e,a)$ plane for a single realization (309 systems). DPs are not shown here. The $t_{\rm GW}=t_{\rm rlx}$ pink line marks the boundary of the GW-dominated region\protect\footnotemark. The tracks of captured systems before they become EMRIs (step 3) are marked in light blue. Systems that cross the EMRI-detection boundary $t_{\rm GW}=10^{-3}t_{\rm rlx}$ (black line) are evolved in step 4 through inspiral until plunge exclusively due to GW emission (dark blue, initially diagonal lines). The $t_{\rm comb} = 100\;\rm Myr$ red curve marks the boundary of the slowly-evolving region. The dashed horizontal line marks the semi-major axis corresponding to an EMRI orbital period of $10^4\;\rm s$, an approximate indicator of entrance into the LISA band. The dotted diagonal line marks pericentres corresponding to the tidal radius $r_{\rm t}$. S-stars are plotted for reference \citep[orbital solutions taken from][]{Gillessen2017, Verberne2025}, as well as the newly observed S301 \citep{S3012026}.

Caption

\emph{Left panel}: Scatter distribution of our captured BHs in the $(a_{\rm cap}-e_{\rm cap})$ parameter space coloured by their final fate (EMRIs: dark brown dots, DPs: pink crosses, SE systems: light orange triangles). Here $r_{\rm BH}=8R_g$ and orbital elements are computed from Newtonian prescriptions. \emph{Right panel}: Fraction of EMRIs as a function of the initial $\beta$. The darker line (circular markers) is the fraction of EMRIs relative to EMRIs and DPs, while the lighter one (star markers) is the fraction of EMRIs to EMRI, DPs, and SE systems. The vertical black line marks $\beta=18.5$, corresponding to $r_{p,\rm{CM}}=r_{\rm BH}$.

Caption

\emph{Left panel}: Probability density function (PDF) of the EMRI eccentricity distribution at plunge. \emph{Right panel}: PDFs of EMRI eccentricity at the threshold of the LISA band. In both panels, the Hills distribution (solid red line) is compared with the \emph{classical}-EMRIs distribution from \citep{Mancieri2026} (black dashed line). The sharp cutoff of the classical distribution at $e^{\rm C}_{\rm plunge}=0.026$ likely results from having excluded the more circular systems that were diffused in the slowly-evolving region of the $(1-e,a)$ plane in \citet{Manciericliff2025}, since they employed a stricter condition to define that region than the one used in this work.

Caption

Probability density function (PDF) of the semi-latus rectum $p$ of the EMRI orbits at the beginning of the evolution with FEW (Step 4). the Hills distribution (solid red line) is compared with the \emph{classical}-EMRIs distribution (data from \citet{Mancieri2026}) (black dashed line). The sharp cutoff in the classical distribution likely arises for the same reason as the cutoff in Fig.~\ref{fig:final ecc}.

Caption

Probability that an EMRI in a given eccentricity bin centred on value $e$, is an Hills EMRI as a function of eccentricity at plunge (\emph{left panel}) and at the entrance of the LISA band (\emph{right panel}). The red upper line is obtained from our initial conditions: assuming the BBHs and single BHs have the same total mass. No further weighting is applied. The other curves are weighted by the product between the relative rate of Hills versus classical EMRIs per galaxy (i.e., $N^{\rm H}_{\rm EMRI}/N^{\rm C}_{\rm EMRI}=157/1344$) and by $\mu/(1-\mu)$. The factor $\mu$ is the fraction of mass that is in binaries vs singles: $\mu=1/2, 1/10,1/100$ means that $50,10,1$ per cent, of the mass is contained in binaries and $50, 90,99$ per cent in singles, respectively. The horizontal lines mark the $5,20,50$ per cent and $95$ per cent probabilities. Vertical dotted lines mark the edges of each bin. The vertical error bars have been computed by propagation of uncertainty as explained in Appendix \ref{appendix}.

Caption

CDF of detected eccentricity combining the two formation channels. \emph{Left panel}: eccentricity at plunge. \emph{Right panel}: eccentricity at the entrance of the LISA band. The red dashed is obtained from our initial conditions: assuming the BBHs and single BHs have the same total mass and differ in number (with BBHs being 1/2 of the single BHs). No further weighting is applied. The other curves are weighted by the relative EMRI rate in the two channels and by the factor $\mu\in[1/2, 1/10,1/100]$, as explained in the caption of Fig. \ref{p(Hills|loge)}. The horizontal lines mark the $20,50$ and $80$ per cent probabilities.

Caption

For each period in a range from $1-500$h, we show the corresponding PDF using a box and whisker representation. The red central line in each box indicates the median of the corresponding distribution, while the box contains the central $50$ per cent (interquartile range, IQR) and the whiskers extend from the lower and upper quartiles to the most extreme values within 1.5 IQR of each quartile. Data points lying beyond these limits are treated as outliers and not plotted. The yellow solid line represent the analytical evolution through Peters equation. The background solid light-blue lines are the trajectories of the various systems starting at different initial periods until plunge. The dotted blue line marks the last stable orbit, i.e. the separatrix. The gray shaded area below it is the loss cone.