Source Parameters
─ ─ Häberle 2024: ≥ 8,200 M☉ lower bound
─ ─ Bañares 2025: ≤ 6,000 M☉ upper bound
Both valid; unreconciled tension preserved.
Physics & citations
Characteristic strain for a circular inspiral (sky/polarisation averaged, Cutler & Flanagan 1994):
h_c(f) = 2√(5/24)/π^{2/3} · c/d · (GM_c/c³)^{5/6} · f^{-1/6}
where M_c = (m₁m₂)^{3/5}/(m₁+m₂)^{1/5} is the chirp mass. Track is cut off at the ISCO frequency.
ISCO frequency (Bardeen 1972, prograde equatorial):
f_ISCO = c³/(πGM) / ((r_ISCO/r_g)^{3/2} + χ)
where χ is the dimensionless spin and r_ISCO/r_g follows from the Kerr ISCO formulae.
Sensitivity curves are analytic approximations (see below). LISA uses the Robson, Cornish & Liu 2019 parameterisation; ET / CE / aLIGO use simplified power-law fits normalised to published design PSDs; DECIGO uses a simplified Yagi & Seto 2011 model; PTA shows approximate SKA-era single-source sensitivity.
Cutler & Flanagan 1994 (characteristic strain) — doi:10.1103/PhysRevD.49.2658
Bardeen, Press & Teukolsky 1972 (Kerr ISCO) — doi:10.1086/151796
Hild et al. 2011 (ET-D sensitivity) — doi:10.1088/0264-9381/28/9/094013
Evans et al. 2021 (Cosmic Explorer) — doi:10.3847/2041-8213/abe938
Häberle et al. 2024, Nature 631:285 — doi:10.1038/s41586-024-07511-z
Bañares-Hernández et al. 2025, A&A 693:A104 — doi:10.1051/0004-6361/202451763