PROTIUM — Parameters

Parameters

PROTIUM v0.5.6

Intrinsic — directly measured at detector · ℳ is detector-frame · independent of cosmological model

ℳ_det Detector-frame chirp mass M☉ · one decimal place (two for ℳ_det < 10 M☉)

The dominant mass parameter governing inspiral frequency evolution, as directly measured by the detector. Equal to (1+z)ℳ_source — cosmological redshift stretches the observed waveform, making the source appear more massive than it is.

ℳ_det = (1+z) × (m₁m₂)^(3/5) / (m₁+m₂)^(1/5)

Source-frame ℳ is a derived quantity: ℳ_source = ℳ_det / (1+z). Recovering it requires importing z from the extrinsic column — the two columns are not fully independent. The fingerprint encodes what an interferometer directly measures, without assuming a cosmological model.

In geometric units (G=c=1): 1 M☉ = GM☉/c³ ≈ 4.9255 μs ≈ 6,996 τ(H). Chirp mass is directly expressible as a geometric flip count — see Mass Verification Protocol.

q Mass ratio dimensionless · range 0–1 · two decimal places

m₂/m₁ where m₂ ≤ m₁. Always ≤ 1 by convention.

χ_eff Effective inspiral spin dimensionless · range −1 to +1 · two decimal places

Mass-weighted aligned spin component. Governs inspiral phase evolution. Distinct from final remnant spin.

Near-zero values are common — do not confuse with a_f, which can be large even when χ_eff ≈ 0.

a_f Final remnant spin dimensionless · range 0–1 · two decimal places

Kerr spin parameter of the merged object. Derived from the post-merger ringdown waveform.

⧖ for BNS — post-merger ringdown above current detector frequency ceiling.

Λ̃ Effective tidal deformability dimensionless · integer

Mass-weighted tidal deformability. Encodes the neutron star equation of state — the physics of matter at nuclear density. Imprinted on the late-inspiral waveform phase.

∅ for BBH — black holes have no tidal deformability by the no-hair theorem.

Extrinsic — positional properties · observer-dependent · encodes sender location

z Cosmological redshift dimensionless · three decimal places

Fractional wavelength increase due to cosmological expansion along the signal path. For multi-messenger events, measurable spectroscopically from the host galaxy. Observer-dependent — two senders at different locations measure different z for the same event. This is the parameter that enables positional reconstruction. z is model-free and universally derivable from spectroscopic observation.

D_L Luminosity distance Mpc · integer

Distance derived from GW waveform amplitude. With z, gives an independent H₀ measurement for multi-messenger events. Retained in Mpc for human adoption.

Universal expression: 1 Mpc = Hl☉^23.165. See the distance ladder for additional landmarks.

Schema Selection

The schema for each event is determined by which parameters are physically meaningful — specifically, the null-state profile of a_f and Λ̃. This reflects the physical nature of the source rather than imposing a classification by mass threshold.

Schema	a_f	Λ̃	Physical basis
BBH	measured or ⧖	∅	No matter surface on either component — tidal deformability is physically absent (no-hair theorem)
NSBH	measured or ⧖	⧖ or constrained	One component has a matter surface, one doesn’t — Λ̃ is asymmetric but physically present; remnant is a Kerr black hole
BNS	⧖	measured or ⧖	Both components have matter surfaces

Duration shortcut: In-band duration provides an initial classifier before full parameter estimation. BBH events typically occupy < ~1 second, BNS events > ~10 seconds, and NSBH events fall in the ~1–10 second intermediate range. The null-state profile is definitive; duration is a practical first-pass.

Mass gap: The boundary between neutron star and black hole is observationally unresolved in the ~2.5–5 M☉ range. Events with components in this mass gap may have ambiguous schema assignment — the null-state profile reflects the physical ambiguity rather than forcing a classification.