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

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) ≈ 2.20 × 10⁻² 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) → r_s = 2 G M / c² as a → 1 (maximally spinning) f_GW increases by up to a factor ~6 at a = 0.998 vs 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)^(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) LISA sensitivity band: ~0.1 mHz to ~0.1 Hz (peak ~3 mHz) OC distance: D = 5.43 kpc = 1.675 × 10²² m

Problems

1. ISCO frequency and LISA band placement

Consider an EMRI with primary M = 8,200 M☉ (Häberle 2024), secondary m₂ = 10 M☉, at OC distance D = 5.43 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 (as assumed in the OCS GW Horizon Plotter), the prograde ISCO radius shrinks to approximately 2.3 rs. By what factor does fGW increase relative to the Schwarzschild case? Is the new frequency still within the LISA band? [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 = 5.43 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 = 1.631×10³⁴ kg f_orb = (2.998×10⁸)³ / (6^1.5 × 2π × 6.674×10⁻¹¹ × 1.631×10³⁴) = 2.694×10²⁵ / (14.697 × 2π × 1.088×10²⁴) = 2.694×10²⁵ / (9.997×10²⁵) = 0.02695 Hz f_GW = 2 × 0.02695 = 0.0539 Hz ≈ 54 mHz 54 mHz is at the high-frequency end of the LISA band (0.1–100 mHz). LISA's peak sensitivity is at ~3 mHz; at 54 mHz LISA's noise floor rises steeply. The signal is technically within the nominal band but near its upper edge.

Part (b): At a = 0.9, r_ISCO ≈ 2.3 r_s (the tool uses the exact Kerr formula). Since f_orb ∝ 1/r_ISCO^(3/2) and r_ISCO shrinks from 6 r_s to 2.3 r_s: f_GW(a=0.9) / f_GW(a=0) = (r_ISCO,0 / r_ISCO,a)^(3/2) = (6/2.3)^1.5 = (2.609)^1.5 = 4.21 So f_GW ≈ 54 × 4.21 ≈ 227 mHz. This is outside the LISA band. A non-spinning 8,200 M☉ IMBH produces EMRI signals near LISA's upper edge; a rapidly spinning one pushes the peak frequency above the band.

Part (c): The tool should show the characteristic strain curve peaking near ~50–200 mHz for these parameters, straddling the LISA noise floor. Students should note whether the peak exceeds the LISA sensitivity curve at 1 yr integration.
2. Characteristic strain and SNR

Use the parameters M = 8,200 M☉, m₂ = 10 M☉, D = 5.43 kpc, a = 0, and Tobs = 4 yr (LISA nominal mission lifetime).

  1. Estimate Ncycles of GW radiation during the last year of inspiral (when the orbit is roughly at the ISCO). Note: at 54 mHz, N_cycles ≈ f_GW × T_obs_1yr. [4 marks]
  2. A rough SNR estimate for LISA detection is SNR ~ hc / Sn(f)^{1/2}, where Sn at 54 mHz is approximately 10−20.5 Hz−1/2. Using the characteristic strain formula, estimate whether the OC EMRI is detectable with SNR > 8 (standard LISA threshold). Assume h₀ ≈ 10−22 as an approximation and compute hc. [6 marks]
  3. How does the detectability change if M = 3,000 M☉ (Baumgardt limit)? Compute the ISCO frequency and comment on detectability qualitatively. [5 marks]
↗ GW Horizon Plotter — M = 3,000 M☉ · a = 0 (Baumgardt limit comparison)
Show solution
Part (a): N_cycles = f_GW × T_1yr = 0.054 Hz × 3.156×10⁷ s = 1.70×10⁶ cycles ~1.7 million GW cycles accumulated in 1 year at the ISCO. This is a monochromatic signal in LISA terms — the frequency evolution (chirp) is slow because the mass ratio is extreme (m₂/M = 10/8200 ≈ 1.2×10⁻³).

Part (b): h_c = h_0 × sqrt(N_cycles) ≈ 10⁻²² × sqrt(1.7×10⁶) = 10⁻²² × 1304 ≈ 1.3×10⁻¹⁹ S_n^(1/2) at 54 mHz ≈ 10⁻²⁰·⁵ = 3.16×10⁻²¹ Hz^(−1/2) For matched-filter SNR we need h_c and the noise spectral density. A rough SNR: SNR ≈ h_c / (S_n^(1/2) × f^(1/2)) ≈ 1.3×10⁻¹⁹ / (3.16×10⁻²¹ × sqrt(0.054)) ≈ 1.3×10⁻¹⁹ / (3.16×10⁻²¹ × 0.232) ≈ 1.3×10⁻¹⁹ / 7.34×10⁻²² ≈ 177 SNR ≈ 177 ≫ 8. Under this rough estimate, the OC EMRI is highly detectable by LISA. However, h₀ ~ 10⁻²² is a generous approximation; students should flag that the actual h₀ depends on the orbital parameters and inclination. The GW Horizon Plotter gives a more precise estimate.

Part (c): For M = 3,000 M☉: f_GW(3000 M☉) = 0.0539 × (8200/3000) = 0.147 Hz 147 mHz is above the nominal LISA band. At this frequency, the LISA noise floor rises rapidly (dominated by laser phase noise). A 3,000 M☉ IMBH would produce an EMRI signal that is marginally above the LISA band, making detection significantly harder. This is an important discriminator: if the IMBH mass is <4,000 M☉, LISA will not detect an EMRI unless spin is modest (a < 0.5). The Häberle mass (8,200 M☉) is actually more favorable for LISA than the Baumgardt upper limit.
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 = 5.43 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 relative to a Gpc source? [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. In a single 4-year LISA mission, there is about a 33% chance of detecting one OC EMRI.

Part (b): With ±1 dex uncertainty: N_expected ranges from ~0.033 to ~3.3 events over 4 yr. In the optimistic case (>3 events), OC is a strong guaranteed LISA source. In the pessimistic case, it's a ~3% probability per mission. The expected value (~0.3 events/mission) puts OC in the "long shot but not negligible" category — comparable to the detection probability of M31-type IMBH sources at ~700 kpc. However, if an EMRI is detected, OC's distance means the SNR would be enormous (see Problem 2c).

Part (c): Characteristic strain h₀ ∝ 1/D_L. Ratio: D_Gpc / D_OC = 1,000 Mpc / (5.43 kpc) = 1,000 × 10³ kpc / 5.43 kpc = 1.84 × 10⁵ h₀(OC) / h₀(Gpc) = D_Gpc / D_OC ≈ 1.84 × 10⁵ OC's proximity makes any EMRI ~2 × 10⁵ times stronger in strain than a typical cosmological EMRI source. An OC EMRI that would be marginally detectable at 1 Gpc would have SNR ~ 10⁵ × 8 ~ 10⁶ at OC's distance — far exceeding LISA's dynamic range. This would require special signal-processing techniques but would be the most precisely measured IMBH mass in history.

References