Co-evolution of Nuclear Star Clusters and Massive Black Holes: Extreme Mass-Ratio Inspirals

TL;DR

This study advances EMRI predictions by integrating GW orbital decay, spin-modified loss cones, and stellar evolution into a self-consistent Monte Carlo framework (GNC) for NSCs hosting mass-growing MBHs. Over 12 Gyr, it reveals how MBH growth from TDEs and stellar mass loss, together with cluster expansion, controls EMRI rates across object types, with SBH-EMRIs predominating and X-MRIs from brown dwarfs becoming detectable only under certain MBH masses. The results show that MBH spin broadens the EMRI-forming parameter space for compact objects and that the MBH growth history significantly impacts EMRI demographics and observable properties in the LISA band, including mass, eccentricity, and inclination distributions. These findings provide time-dependent, astrophysically realistic benchmarks for interpreting future space-based GW observations and highlight how NSC evolution shapes the EMRI landscape across cosmic time.

Abstract

We explore extreme mass-ratio inspirals (EMRIs) in the co-evolution of massive black holes (MBHs) and nuclear star clusters (NSCs), which host diverse stellar populations across a wide range of masses. The dynamics are simulated self-consistently with GNC, which we have updated to incorporate gravitational wave orbital decay, the loss cone of a spinning MBH, and stellar evolution. Over $12$ Gyr, we investigate the evolution of the NSC with a mass-growing MBH, as well as the EMRIs of stellar black holes, neutron stars, white dwarfs, brown dwarfs (BDs), and low-mass main-sequence stars (MSs), along with tidal disruption events (TDEs) involving MSs, BDs, and post-MSs. The mass growth of the MBH contributed by TDEs is typically $\sim 10^7\,M_{\odot}$, $\sim 10^6\,M_{\odot}$, and $\sim 5\times10^4\,M_{\odot}$ for massive, Milky-Way-like, and smaller NSCs, respectively. Between $40\%$ and $70\%$ of the stellar mass is lost during stellar evolution, which dominates the mass growth of the MBH if a significant fraction of the lost mass is accreted. The evolution of EMRI rates is generally affected by the cluster's size expansion or contraction, stellar population evolution, MBH mass growth, and the stellar initial mass function. The EMRI rates for compact objects peak at early epochs ($\lesssim 1$ Gyr) and then gradually decline over cosmic time. LISA-band ($0.1$ mHz) EMRIs involving compact objects around Milky-Way-like MBHs tend to have high eccentricities, while those around spinning MBHs preferentially occupy low-inclination (prograde) orbits. In contrast, MS- and BD-EMRIs usually have eccentricity and inclination distributions that are distinct from those of compact objects.

Figures (15)

• Figure 1: The mass-radius relation for BDs ($m_\star=0.01M_{\odot}\sim 0.1M_{\odot}$) and low-mass MSs ($0.1M_{\odot}\sim 1M_{\odot}$). Black lines are results from MOBSE, while colored lines are from 2003AA...402..701B. The green dashed lines show the stellar radius at which the tidal radius of the star, $r_{\rm td}$, equals the pericenter of the ISO ($r_{\rm ISO}$). For prograde (retrograde) orbits around a maximally spinning MBH, $r_{\rm ISO}=2.12r_g$ ($r_{\rm ISO}=11.65r_g$). For a Schwarzschild MBH, $r_{\rm ISO}=8r_g$.
• Figure 2: Examples of evolutionary trajectories of SBHs in model M2 (see Table \ref{['tab:modelnoSE']}) simulated by GNC, plotted in the space of energy, angular momentum, and pericenter distance. Thin lines ending with filled red circles mark plunge events, while those ending with filled red squares mark EMRI events. Top left panel: The energy and angular momentum evolution for a maximally spinning MBH. $x=E/\sigma_0^2$ and $j=J/J_c(x)$ are the dimensionless energy and angular momentum (see text in Section \ref{['sec:method']} for their definitions). The loss cone size depends on the amplitude ($a$) and inclination angle ($i$) of the MBH's spin (defined in Equation \ref{['eq:spin_inc']}). Top right panel: Similar to the left panel but for the evolution in the space of energy and pericenter distance. $r_0=3.1$ pc is a characteristic distance. Bottom panels are similar to the top panels but for the case of a Schwarzschild MBH ($a=0$). The cyan filled triangle, star, and square in each panel mark the intersection between $T_{\rm GW}=0.1T_{\rm rlx}$ and the loss cone for $a=1$, $a=0$, and $a=-1$ (equivalent to $a=0$ with $i=180^\circ$), respectively.
• Figure 3: Critical boundaries in the space of dimensionless energy $x$ and angular momentum $j$ for different stellar objects in Model M5. In all panels, black solid lines correspond to $T_{\rm GW} =0.1T_{\rm rlx}$, where $T_{\rm rlx}$ and $T_{\rm GW}$ are the relaxation and GW timescales, respectively (defined in Equation \ref{['eq:emri_cri']}). Red, blue, and green dashed lines indicate the pericenter of the ISO: $r_{p} =r_{\rm ISO}=2.12r_g$ (for $a=1$ and $i=0^\circ$), $r_{\rm ISO}=8r_g$ (for $a=0$), and $r_{\rm ISO}=11.6r_g$ (for $a=1$ and $i=180^\circ$), respectively. The black dashed line in the right panel represents the tidal radius of a BD ($r_{p}=r_{\rm td}=4r_g$) for an MBH mass of $4\times10^6M_{\odot}$. The loss cone size is given by $r_{p,\rm lc}={\rm max}(r_{\rm ISO},r_{\rm td})$, and its intersection with $T_{\rm GW}=0.1T_{\rm rlx}$ defines the critical distance $a_{\rm crit}$ (marked by red, blue, green, or black filled circles in each panel). Horizontal dash-dotted lines, colored the same as the filled circles, show the corresponding $a_{\rm crit}$ given by Equation \ref{['eq:emri_cir_ana']}.
• Figure 4: Evolution of the EMRI and plunge event rates in five-component models (model M5 or M5_2 in Table \ref{['tab:modelnoSE']}), with a Schwarzschild ($a=0$) or maximally spinning MBH ($a=1$). In these simulations, the MBH mass is fixed, and stellar evolution is ignored.
• Figure 5: Left panel: Evolution in Model M5G82 (see Table \ref{['tab:modelnoSE']}) of the MBH mass (black solid line), the theoretical mass growth under the Eddington limit (black dashed line), the current mass of the gas reservoir ("Gas reservoir (left)"), and the cumulative contributions from TDEs of stars ("TD (stars)"), SBH-EMRIs ("EMRI (SBHs)"), and plunge events of SBHs and WDs ("Plunge (SBHs)" and "Plunge (WDs)"). Contributions from plunge/EMRI events of both NSs and BDs are negligible and are thus not shown for clarity. Right panel: Evolution in the same model of the event rates of TDEs of stars or BDs, EMRIs, and plunge events of compact objects.