Ellipticity upper limits from O4/O5 LIGO and Einstein Telescope for rotating pulsars in Omega Centauri's core — the CW complement to the LISA inspiral channel
A rotating neutron star with a non-zero ellipticity ε = (I_xx − I_yy) / I_zz emits continuous GWs at twice the spin frequency. The strain amplitude reaching Earth at distance d is:
h₀ = (4π²G / c⁴) × (I_NS × ε × f_gw²) / d
where I_NS = 10³⁸ kg·m² is the canonical NS moment of inertia, f_gw = 2 × f_spin, and d = 5.49 kpc (Omega Centauri distance, Baumgardt & Hilker 2018).
For a coherent F-statistic search of duration T_obs, the 95% CL detection threshold is approximately:
h_thresh ≈ 11.4 × √(S_h(f) / T_obs)
where S_h(f) is the one-sided power spectral density (units Hz⁻¹) and the factor 11.4 encodes the standard signal-to-noise threshold for the chi-squared F-statistic (Prix 2009). Sensitivity improves as √T_obs — halving the threshold requires 4× the observing time.
O4 (simplified analytic): √S_h = 3.5×10⁻²⁴ × √(1 + (100/f)⁴ + (f/600)²) h/√Hz — captures the seismic wall below ~30 Hz, shot-noise rise above ~300 Hz, and peak sensitivity near 100–150 Hz.
O5: 2× better than O4 in strain amplitude (√S_h_O5 = 0.5 × √S_h_O4), consistent with the LIGO O5 design sensitivity document.
Einstein Telescope (ET-D): √S_h = 1.5×10⁻²⁴ × √(1 + (15/f)⁶ + (f/1500)²) — extends low-frequency reach to ~5 Hz via underground siting and cryogenic mirrors (Punturo et al. 2010).
The spin-down limit ε_SD is the maximum ellipticity consistent with the observed spin-down rate being entirely due to GW emission. For an Omega Centauri millisecond pulsar (typical P ~ 3 ms, Ṗ ~ 10⁻²⁰), the estimate is:
ε_SD ≈ 10⁻⁸ × (100/f_gw)^(5/2)
MSPs typically have ε_SD ~ 10⁻⁹–10⁻⁸, well below the crust-yield limit (~10⁻⁷). Reaching ε_SD sensitivity requires years of coherent integration, which becomes computationally expensive for wide-band all-sky searches but is tractable for targeted searches with known ephemerides.
If Omega Centauri contains a dark cluster of N unseen black holes or neutron stars, each contributing a CW signal independently, the total incoherent power scales as N × h₀². Coherent combination across N sources is not feasible, but stacking raises the effective amplitude threshold. The displayed N is the number of identical ε emitters required so that one signal crosses the detection threshold (i.e., SNR_single = h₀/h_thresh > 1/√N for stacking purposes — simplified here to the number where SNR reaches 1).
LIGO O4 data (from May 2023) is released publicly via the Gravitational Wave Open Science Center (GWOSC, gwosc.org) on a rolling basis. A targeted CW search on known OC pulsars from the ATNF catalog requires only the ephemerides and a few days of compute on a GPU cluster. No new telescope time is needed.