Current IMBH constraints (Tools 2, 9, 13) are all static — they say nothing dynamic is happening now. LISA (launch ~2035) would catch a stellar-mass compact object spiralling into the IMBH and produce a clean dynamical signature. This tool computes the characteristic-strain trajectory and SNR for that scenario.
For two masses M (primary BH) and m (compact companion) on a circular orbit at orbital frequency f_orb, the dominant gravitational-wave frequency is f_GW = 2 f_orb. Energy is radiated according to the Peters (1964) formula, and the orbit shrinks. The characteristic strain — the strain amplitude integrated over the number of cycles spent near each frequency — is
h_c(f) = √(2/3) · π⁻²/³ · (G M_chirp / c³)⁵/⁶ · f⁻¹/⁶ · c / d_L
where M_chirp = (M·m)³/⁵/(M+m)¹/⁵ is the chirp mass and d_L is the luminosity distance. This is the slow-rising line on the strain plot. The signal stops at f_ISCO ≈ c³/(6√6·π·G·M_BH) = 4.4 kHz × (M☉/M_BH) — the innermost stable circular orbit, beyond which the inspiral plunges into the horizon and the perturbative waveform breaks down. For an 8,200 M☉ IMBH, f_ISCO ≈ 0.54 Hz, well inside LISA's sensitive band.
The purple curve uses the Robson, Cornish & Liu (2019) sky-averaged analytical LISA strain sensitivity, the standard parameterisation for back-of-envelope SNR estimates. The bucket bottoms out around f ≈ 5 mHz with strain sensitivity h_n ≈ 4×10⁻²¹ in a one-year integration. Outside the bucket — below ~10⁻⁴ Hz acceleration noise dominates; above ~0.1 Hz optical metrology noise dominates — sensitivity falls steeply.
The matched-filter SNR over the chirp from f_start to f_ISCO is
SNR² = ∫ (h_c(f)/h_n(f))² · (df/f)
Numerically integrated by the tool. Conventional "detection threshold" for LISA is SNR ≥ 10. Above that, the signal is detectable; above ~20 the source parameters (mass, spin, location, distance) can be estimated to scientific precision. For the OC + stellar-BH default, SNR is typically in the few-dozen to few-hundred range depending on inclination and chirp mass, well above threshold.
The harder question is the event rate. arXiv:2501.13466 (Jan 2025) ran 100 MOCCA simulations of IMRI formation in globular clusters and found that of the IMRIs that form, less than 10% have SNR above the LISA threshold — but a population of ~10⁴ globular clusters in the universe could produce a few-per-decade rate of detectable events. For Omega Centauri specifically, the rate is dominated by the dynamical interaction cross-section in the dense core and is highly uncertain (a single dynamical event in the past 13 Gyr is plausible). The "rate per Hubble time per cluster" row gives the order-of-magnitude expectation; multiply by the number of clusters and the mission lifetime to get an expected detection count.
The current IMBH constraints are all upper bounds (Bañares, JWST, TRAPUM) plus one disputed lower bound (Häberle). Even if all parties agree that "a 6,000–10,000 M☉ IMBH is allowed in OC," the question of whether it actually exists is observationally answerable only by either (a) a future tighter kinematic measurement, (b) a future tighter pulsar timing measurement (SKA), or (c) a gravitational-wave detection. LISA gives the cleanest "yes/no" because it sees the dynamical signature of a compact orbit, not the time-averaged enclosed mass profile. This tool lets the visitor see exactly what region of (M_BH, m_companion, distance) LISA can cover.
Advanced undergrad: derive the characteristic strain formula from the quadrupole radiation formula; reproduce the f⁻¹/⁶ slope. Graduate: integrate SNR over the chirp; explain why the bucket frequency dominates the SNR budget; compute the parameter-estimation precision from Fisher matrix arguments. Project seed: compute the expected number of OC-like IMRIs in a 4-year LISA mission using the MOCCA-style rate and compare to the IMBH constraints from constraint-stacker.
The Laser Interferometer Space Antenna (LISA) was adopted by ESA in January 2024 as an L-class mission with NASA partnership, targeting launch in 2035 (full operations from ~2037). Three spacecraft in heliocentric orbit form a 2.5-million-km triangular interferometer, sensitive to gravitational waves in the 10⁻⁴ to 10⁻¹ Hz band — exactly the regime where extreme- and intermediate-mass-ratio inspirals (EMRIs / IMRIs) emit. The LISA Pathfinder demonstration mission (2015–2017) successfully validated the technology.
An EMRI is a stellar-mass compact object (typically a stellar BH, 10–60 M☉) inspiralling into a supermassive BH (10⁵–10⁷ M☉). Predicted detection rates: ~10–100 EMRIs/year at LISA design sensitivity (Babak et al. 2017 PRD 95:103012), with significant uncertainty from the M_BH mass function and capture-rate physics. An IMRI involves an IMBH (10²–10⁵ M☉) as either the central or inspiralling object. IMRI rates are much more uncertain — anywhere from ~0.1 to ~10 per year depending on whether IMBHs are common in GCs.
A Häberle-mass (~8200 M☉) IMBH at OC's distance (5.4 kpc) would produce IMRI signals from any stellar-mass compact object that gets captured into a tight orbit. LISA's strain sensitivity at 10⁻³ Hz is ~10⁻²⁰ Hz^(−1/2); a 10 M☉ BH inspiralling into the OC IMBH would produce a coherent signal detectable for 1–10 years before merger. Even one detection from OC would settle the IMBH question definitively, with mass and spin measurements at 1% precision. The challenge is rate — capture timescales for the few stellar BHs expected in OC are uncertain.