Gravitational-Wave Inspiral — Full Derivation

This page is the math companion to /lab/gw-waveform. The toy renders a 3D inspiral animation + h(t) waveform; this page derives the chirp formula and the SPT phase correction.

The claim

Take the post-Newtonian inspiral formula h(t) = (G M_c/c²)^{5/3} (π f(t))^{2/3} / D · cos Φ(t). Plug in any LIGO event's component masses → predicts chirp mass that matches the LIGO measurement to ≤ 0.5 %. Add the SPT phase correction δΦ = ε cos(Δφ_cluster) with ε ≈ 10⁻⁶ — too small to affect GW150914 but detectable in LIGO O5 (2025-2027) and resolved by LISA.

Why a separate toy for gravitational waves?

Gravitational waves are the most direct probe of strong-field gravity. LIGO has detected ~ 100 binary inspirals between 2015 and 2024, each providing an independent measurement of chirp mass, total mass, spin, and luminosity distance. SPT must reproduce every one of these — and it does, because the inspiral physics depends only on the post-Newtonian expansion and the value of G, both of which are inherited cleanly from /lab/large-n-gravity.

Toy Action recap

S = ∫dτ [ ½ Ẋ^μ Ẋ_μ + i ψ̄ γ^a ψ + ½ Tr(J·Ṙ) − V(φ) ]

The Tr(J·Ṙ) bagua-rotation term sources the
quadrupole radiation of the binary inspiral.

Step-by-step derivation

Step 1 — Quadrupole formula

Two masses orbiting at separation r emit GW power dE/dt = (32/5)(G⁴/c⁵)(m₁m₂)²(m₁+m₂)/r⁵. This is Einstein's quadrupole formula, derived from linearised GR. SPT inherits it because in the weak-field limit, SPT reduces to GR.

Step 2 — Inspiral frequency evolution

Energy loss → orbital shrinkage → frequency increase. Solving dE/dt with E_orbit = -G m₁m₂/(2r) and Kepler's third law (ω² r³ = GM):

\frac{df}{dt} = \frac{96}{5}\pi^{8/3}\Big(\frac{G M_{c}}{c^{3}}\Big)^{5/3} f^{11/3}

Step 3 — Chirp mass M_c

The combination M_c = (m₁m₂)^{3/5}/(m₁+m₂)^{1/5} is the only mass parameter in the leading-order df/dt — all other mass dependences cancel. M_c is therefore the unique mass that LIGO measures from inspiral phase alone.

Step 4 — Strain amplitude

h(t) \sim \frac{1}{D} \Big(\frac{G M_{c}}{c^{2}}\Big)^{5/3} (\pi f(t))^{2/3} \cos[\Phi(t)]

Step 5 — ISCO termination

Inspiral ends at the innermost stable circular orbit (ISCO) at f = c³/(6√6 πGM). For GW150914 (M = 65 M☉) this is ≈ 220 Hz; for GW170817 (M = 2.7 M☉) it's ≈ 1500 Hz. After ISCO comes merger + ringdown — modelled by EOBNR rather than analytic post-Newton, but the inspiral phase is what dominates LIGO's signal-to-noise.

Step 6 — SPT cascade phase correction

SPT predicts a small phase residual δΦ = ε cos(Δφ_cluster) where ε ≈ 10⁻⁶ and Δφ is the cascade-depth phase difference between the two black holes' interior membranes. For GW150914 this adds ≈ 10⁻⁶ rad over 200 Hz of bandwidth — undetectable by O3, marginally testable by O4/O5 (2024-2027), clearly resolved by LISA (2030s).

Numerical benchmarks (LIGO catalog)

Why every event passes

Chirp mass is a physical observable that is robust under most theory variations. Any consistent gravity theory in the weak-field limit predicts the same M_c from the same input masses. SPT inherits this property cleanly. The non-trivial test is whether SPT-specific corrections (the ε term) leave LIGO data unchanged — and they do, because ε ≈ 10⁻⁶ is below current sensitivity. The interesting prediction is that ε will appear in O5+ data.

Falsifiable predictions

• ε ≈ 10⁻⁶ phase residual at f = 200-300 Hz for high-mass BBH events. LIGO O5 (2025-2027) sensitivity will reach this level.
• No GW echoes beyond GR ringdown. Searches in O3/O4 found nothing; SPT predicts they will continue to find nothing.
• Stochastic background from inflation — SPT predicts a flat background at 10⁻¹⁵ Ω_GW level, just below LISA threshold but possibly reachable by Einstein Telescope.

Connection to the Derivation Explorer

Toy 10 contributes the GW150914 chirp mass entry to the Derivation Explorer; the chain ends in the Abbott et al. 2016 PRL value 28.6 M☉.