Graduate Seminar: Compact Objects in Dense Stellar Systems
Omega Centauri Society Problem Sets

Problem Set 4 — EMRI Gravitational Wave Detectability

Estimated time: 3–4 hours Total marks: 40 Tools: gw-horizon-plotter omegacentauri.me/pset-4-emri.html
Learning objectives
  1. Compute the ISCO orbital frequency and gravitational wave frequency for an EMRI in the Omega Centauri IMBH
  2. Derive the characteristic strain hc and assess LISA detectability
  3. Understand how IMBH mass and spin (Kerr parameter a) shift the GW frequency into or out of the LISA band
  4. Estimate the EMRI rate in OC and the expected number of LISA-detectable events during mission lifetime
Adopted values (used throughout PS 1–4)
  1. Distance to OC: d = 5.49 kpc = 1.694 × 1020 m (oMEGACat VI, arXiv:2503.04903).
  2. Häberle mass: M ≥ 8,200 M☉, a lower bound on the enclosed mass. Baumgardt: M < 3,000 M☉, a 3σ upper limit. Problems 1–3 treat 8,200 M☉ as a working value, and a larger mass moves every frequency below down in proportion.
  3. LISA band: 0.1 mHz to 100 mHz, peak sensitivity near 3 mHz (Robson, Cornish & Liu 2019). Nominal mission Tobs = 4 yr.
  4. Constants: G = 6.674 × 10−11, c = 2.998 × 108 m/s, M☉ = 1.989 × 1030 kg, 1 yr = 3.156 × 107 s.

Background

An extreme mass-ratio inspiral (EMRI) occurs when a stellar-mass compact object (neutron star or black hole, m₂ ~ 1–30 M☉) slowly spirals into a massive black hole (M₁ ≫ m₂) due to gravitational wave emission. EMRIs are a primary science target for LISA (Laser Interferometer Space Antenna), sensitive in the mHz frequency band.

For a circular Schwarzschild orbit at the ISCO, the orbital frequency and GW frequency are:

ISCO radius (Schwarzschild, a=0): r_ISCO = 6 G M / c² = 6 r_s / 2 = 3 r_s ISCO orbital frequency: f_orb = c³ / (6^(3/2) · 2π · G M) = 2.198 × 10³ Hz × (M☉ / M) GW frequency (l=m=2 dominant mode): f_GW = 2 × f_orb For Kerr (spin a), the prograde ISCO shrinks: r_ISCO(a) → G M / c² as a → 1 (maximally spinning) i.e. from 6 GM/c² at a = 0 down to 1 GM/c² The ISCO frequency is NOT (r_ISCO)^(-3/2) in Kerr. The correct orbital frequency, in units where r is measured in r_g = GM/c², is Ω(a) = (c³ / G M) / (r_ISCO^(3/2) + a) (prograde) so the frequency gain from spin must be read off that expression, not from a radius ratio. It is a factor 3.31 at a = 0.9 and 6.19 at a = 0.998, relative to a = 0. Characteristic strain (Cutler & Flanagan 1994): h_c = h_0 × sqrt(N_cycles) h_0 ≈ (4/√5) × (G μ / c²) × (G M / c²)^(1/3) × (π f_GW / c)^(2/3) / D_L μ = m₁ m₂ / (m₁ + m₂) (reduced mass, ≈ m₂ for EMRI) D_L = luminosity distance N_cycles ≈ f_GW × T_obs (number of GW cycles in T_obs) (The 1/c inside the (π f_GW / c)^(2/3) factor is what makes h_0 dimensionless; dropping it changes the answer by ~13 orders.) LISA sensitivity band: ~0.1 mHz to ~100 mHz (peak ~3 mHz) OC distance: D = 5.49 kpc = 1.694 × 10²⁰ m

Problems

1. ISCO frequency and LISA band placement

Consider an EMRI with primary M = 8,200 M☉ (the Häberle 2024 lower bound, used here as a working value), secondary m₂ = 10 M☉, at OC distance D = 5.49 kpc.

  1. Compute fGW at the Schwarzschild ISCO (a = 0). Is this frequency within the LISA sensitivity band (0.1 mHz – 100 mHz)? [6 marks]
  2. For a rapidly spinning IMBH with a = 0.9, the prograde ISCO radius shrinks to 2.32 GM/c² (from 6 GM/c² at a = 0). Using the exact Kerr orbital frequency in the Background, by what factor does fGW increase relative to the Schwarzschild case? Why is that factor not (6/2.32)3/2? [5 marks]
  3. Use the GW Horizon Plotter (pre-filled link below) to verify your fGW estimate. Does the tool's LISA sensitivity curve overlap the computed peak frequency? [4 marks]
↗ GW Horizon Plotter — M = 8,200 M☉ · m₂ = 10 M☉ · d = 5.49 kpc · a = 0.9
Show solution
Part (a): f_GW(Schw) = 2 × f_orb = 2 × c³ / (6^(3/2) · 2π · G M) Constants: G = 6.674×10⁻¹¹, c = 2.998×10⁸ m/s, M☉ = 1.989×10³⁰ kg M = 8200 × 1.989×10³⁰ = 1.6310×10³⁴ kg G M = 1.0885×10²⁴ m³/s² f_orb = (2.998×10⁸)³ / (6^1.5 × 2π × 1.0885×10²⁴) = 2.6946×10²⁵ / (14.697 × 6.2832 × 1.0885×10²⁴) = 2.6946×10²⁵ / 1.0052×10²⁶ = 0.2681 Hz f_GW = 2 × 0.2681 = 0.5362 Hz = 536 mHz Check against the constant in the Background: 2,198 Hz / 8,200 = 0.2681 Hz, the same number.

536 mHz is above the LISA band, whose upper edge is 100 mHz, by a factor of 5.4. An 8,200 M☉ black hole is light enough that its ISCO frequency sits past LISA's high-frequency cutoff, where the noise floor is already rising as f². That does not put the source out of reach: the ISCO marks the end of the inspiral, and the system radiates at every lower frequency on its way there. The signal crosses the whole LISA band before merging. Only the final, loudest part of the waveform lies above it.

Part (b): The naive answer, (6/2.32)3/2 = 4.16, applies to a Newtonian or Schwarzschild orbit where Ω ∝ r−3/2. In Kerr the prograde orbital frequency carries the frame-dragging term in the denominator: Ω(a) = (c³/GM) / (r_ISCO^(3/2) + a) [r in units of GM/c²] a = 0: r = 6.000, r^1.5 + a = 14.697 + 0 = 14.697 a = 0.9: r = 2.321, r^1.5 + a = 3.536 + 0.9 = 4.436 f_GW(a=0.9) / f_GW(a=0) = 14.697 / 4.436 = 3.31 So fGW ≈ 536 × 3.31 = 1,776 mHz = 1.78 Hz, further above the LISA band still. The same expression at a = 0.998 gives r = 1.237 and a ratio of 6.19, which is where the "factor ~6" in the Background comes from. The radius-ratio shortcut overstates the a = 0.9 gain by 26 per cent because it ignores the +a term.

Part (c): The tool reports f_ISCO = 0.535 Hz at a = 0 and 1.774 Hz at a = 0.9. The small offset from the hand values (0.536 and 1.776 Hz) is because the tool uses the total mass m₁ + m₂ = 8,210 M☉ rather than the primary alone. Its LISA sensitivity curve stops at 100 mHz and the strain track continues past it to the ISCO tick, so the peak does not overlap the LISA curve at all; the detectable portion of the track is the segment below 100 mHz, in the inspiral rather than at the ISCO.
2. Characteristic strain and SNR

Use the parameters M = 8,200 M☉, m₂ = 10 M☉, D = 5.49 kpc, a = 0, and Tobs = 1 yr.

  1. Estimate Ncycles of GW radiation during the last year of inspiral, when the orbit is close to the ISCO: N_cycles ≈ f_GW × T_obs. [4 marks]
  2. Compute h₀ from the Background formula, then hc = h₀ √N. A monochromatic-source SNR estimate is SNR ≈ hc / (Sn(f)1/2 × f1/2). At 536 mHz the Robson, Cornish & Liu (2019) LISA fit gives Sn1/2 = 2.42 × 10−19 Hz−1/2. Is the OC EMRI detectable at SNR > 8? [6 marks]
  3. How does this change if M = 3,000 M☉ (Baumgardt limit)? Compute the ISCO frequency and comment on detectability. [5 marks]
↗ GW Horizon Plotter — M = 3,000 M☉ · a = 0 (Baumgardt limit comparison)
Show solution
Part (a): N_cycles = f_GW × T_1yr = 0.5362 Hz × 3.156×10⁷ s = 1.692×10⁷ cycles About 17 million GW cycles in one year at the ISCO. The signal is nearly monochromatic over the observation: the frequency evolution is slow because the mass ratio is extreme (m₂/M = 10/8200 = 1.2 × 10⁻³).

Part (b): μ ≈ m₂ = 10 M☉ = 1.989×10³¹ kg G μ / c² = 1.3275×10²¹ / 8.988×10¹⁶ = 1.4769×10⁴ m G M / c² = 1.0885×10²⁴ / 8.988×10¹⁶ = 1.2111×10⁷ m (G M/c²)^(1/3) = 229.6 m (π f_GW / c)^(2/3) = (1.6846 / 2.998×10⁸)^(2/3) = (5.619×10⁻⁹)^(2/3) = 3.162×10⁻⁶ m^(−2/3) 4/√5 = 1.7889 D_L = 5.49 kpc = 1.6942×10²⁰ m h_0 = 1.7889 × 1.4769×10⁴ × 229.6 × 3.162×10⁻⁶ / 1.6942×10²⁰ = 19.18 / 1.6942×10²⁰ = 1.13×10⁻¹⁹ h_c = h_0 × sqrt(N) = 1.13×10⁻¹⁹ × sqrt(1.692×10⁷) = 1.13×10⁻¹⁹ × 4114 = 4.66×10⁻¹⁶ SNR ≈ h_c / (S_n^(1/2) × f^(1/2)) = 4.66×10⁻¹⁶ / (2.42×10⁻¹⁹ × sqrt(0.5362)) = 4.66×10⁻¹⁶ / (2.42×10⁻¹⁹ × 0.7323) = 4.66×10⁻¹⁶ / 1.772×10⁻¹⁹ ≈ 2.6×10³ SNR ≈ 2,600, far above the threshold of 8, and that is after paying the noise penalty for sitting at 536 mHz, where Sn1/2 is roughly 80× worse than at LISA's 3 mHz minimum. The reason is distance: OC is five orders of magnitude closer than a cosmological EMRI host (Problem 3c), and h₀ ∝ 1/DL. Two caveats belong on this number. It assumes a full year of coherent integration on a source whose ISCO frequency is above the band LISA is specified for, so the practical SNR is set by the part of the inspiral track that falls below 100 mHz rather than by the ISCO value used here. And the h₀ expression is a sky- and inclination-averaged leading-order estimate; a real analysis carries orbital eccentricity, inclination and the detector response.

Part (c): For M = 3,000 M☉: f_GW(3000 M☉) = 0.5362 × (8200/3000) = 1.466 Hz A lighter black hole has a smaller ISCO radius and therefore a higher ISCO frequency: 1.47 Hz, nearly 15× above the LISA band edge, against 5.4× for the 8,200 M☉ case. At that frequency Sn1/2 has risen to 6.6 × 10⁻¹⁹ Hz−1/2, another factor of 2.7 worse. The inspiral still passes through the LISA band on its way up, so the source is not lost, but a larger fraction of the loud final phase falls outside the instrument. The mass dependence runs the same way throughout: heavier central masses put more of the waveform inside the LISA band, so the Häberle lower bound is the more favourable of the two hypotheses for LISA, and a mass well above 8,200 M☉ would be better still.
3. EMRI rate in Omega Centauri

The EMRI rate in a globular cluster scales roughly as:

Γ_EMRI ≈ A × (M_IMBH / 10⁴ M☉) × (n_* / 10⁵ pc⁻³) where: A ≈ 0.1 events/yr (normalization; uncertain to ~1 dex) n_* = stellar density in the loss cone ≈ 10⁵ pc⁻³ for OC's core (Hopman & Alexander 2005; Amaro-Seoane 2018)
  1. Estimate ΓEMRI for OC using M = 8,200 M☉. Over LISA's 4-year mission lifetime, how many EMRIs from OC would you expect? [5 marks]
  2. The order-of-magnitude uncertainty on A means the true rate could be 10× higher or lower. What range of expected EMRIs does this imply? Is OC a reliable LISA source or a long shot? [5 marks]
  3. OC is at D = 5.49 kpc, while typical LISA EMRI science targets are galaxies at D ~ 1 Gpc. By what factor does OC's proximity increase the characteristic strain h₀ of the same intrinsic source relative to a Gpc distance? [5 marks]
Show solution
Part (a): Γ_EMRI ≈ 0.1 × (8200 / 10000) × (10⁵ / 10⁵) = 0.1 × 0.82 × 1.0 ≈ 0.082 events/yr Over 4 yr: N_expected = 0.082 × 4 ≈ 0.33 events So we expect roughly 1 event per 3 missions. For a Poisson process with mean 0.33, the probability of at least one event in a single 4-year mission is P(≥1) = 1 − e−0.33 = 0.28, or 28%. (The expected number and the probability of at least one differ once the mean is not small; quoting 33% for both conflates them.)

Part (b): With ±1 dex uncertainty, N_expected ranges from 0.033 to 3.3 events over 4 yr, so P(≥1) runs from 1 − e−0.033 = 3.2% to 1 − e−3.3 = 96%. The optimistic end makes OC a near-certain source; the pessimistic end makes it a 3% shot. The central value of ~0.33 events per mission, at 28%, is a long shot rather than a guaranteed source, and the dominant uncertainty is the rate normalisation A rather than anything about OC. If an EMRI is detected, its SNR would be very large (Problem 2b).

Part (c): Characteristic strain h₀ ∝ 1/D_L. Ratio: D_Gpc / D_OC = 1 Gpc / 5.49 kpc = 1×10⁹ pc / 5.49×10³ pc = 1.82 × 10⁵ h₀(OC) / h₀(Gpc) = D_Gpc / D_OC = 1.82 × 10⁵ Moving one and the same source from 1 Gpc to OC's distance multiplies its strain by 1.8 × 105. A source at SNR 8 at 1 Gpc would reach SNR ≈ 1.5 × 106 at 5.49 kpc. The comparison is a scaling exercise rather than a like-for-like one: LISA's cosmological EMRI targets are ~106 M☉ massive black holes whose ISCO frequencies fall near the band centre, whereas the OC primary is 102–103 times lighter and radiates above the band at merger, which is why the SNR computed in Problem 2b is ~2.6 × 103 rather than 106. What survives the caveat is the direction of the effect: proximity is by far the largest single factor working in OC's favour as a LISA target.

References