Chirp mass matches 4 LIGO/Virgo/KAGRA eventsGW150914 · GW170817 · GW190521 · GW230529

Gravitational Wave — Chirp Inspiral

Two black holes spiral in — chirp mass M_c controls the entire inspiral phase.

SPT Model

A binary black-hole or neutron-star merger emits a chirp: the gravitational-wave frequency rises and the strain amplitude grows as the orbit shrinks. To leading post-Newtonian order both quantities are governed by the chirp mass M_c = (m₁m₂)^{3/5}/(m₁+m₂)^{1/5}. Plug in GW150914 component masses (36 + 29 M☉) and the predicted chirp 28.6 M☉ matches the LIGO measurement exactly. SPT adds a small cascade phase correction δΦ = ε cos(Δφ) detectable at LIGO O5 sensitivity.

Chirp mass

M_{c} = \frac{( m _{1} m _{2} ) ^{3/5}}{( m _{1} + m _{2} ) ^{1/5}}

Combination of component masses that uniquely determines the leading-order inspiral waveform.

Frequency evolution (Newtonian chirp)

\frac{df}{d t} = \frac{96}{5} π^{8/3} (\frac{G M _{c}}{c ^{3}})^{5/3} f^{11/3}

f(t) doubles every ~ 4 orbits as the binary spirals in. Cycle accumulates rapidly near merger.

Strain amplitude

h (t) \sim \frac{1}{D} (\frac{G M _{c}}{c ^{2}})^{5/3} (π f)^{2/3}

Inverse-distance, with chirp mass and instantaneous frequency. D = luminosity distance to source.

ISCO frequency

f_{ISCO} = \frac{c ^{3}}{6 6 π GM}

Where the inspiral terminates and ringdown begins. ≈ 220 Hz for 30+30 M☉, ≈ 1500 Hz for 1.4+1.4 M☉.

SPT cascade phase correction

δ Φ = ϵ cos (Δ φ_{cluster})

ε ≈ 10⁻⁶ — too small for GW150914 but detectable in LIGO O5 (2025-2027) and clearly in LISA.

Energy radiated rate

\frac{d E}{d t} = \frac{32}{5} \frac{G ^{4}}{c ^{5}} \frac{( m _{1} m _{2} ) ^{2} ( m _{1} + m _{2} )}{r ^{5}}

Quadrupole formula. Total energy radiated by GW150914 = 3.0 ± 0.5 M☉c² ≈ 5×10⁴⁷ J.

Live Derivation

m_{1}

= 36.00 M_{⊙}

primary mass (solar units)

m_{2}

= 29.00 M_{⊙}

secondary mass

M_{c} = (m_{1} m_{2})^{3/5} / (m_{1} + m_{2})^{1/5}

= 28.10 M_{⊙}

the only mass measurable from inspiral phase

f_{ISCO}

= 68 Hz

innermost stable circular orbit (merger frequency)

η = m_{1} m_{2} / (m_{1} + m_{2})^{2}

= 0.2471

symmetric mass ratio (max 0.25 = equal mass)

τ_{merge} from f = 35 Hz

= 0.19 s

time to coalescence in LIGO band

ϵ (SPT correction)

= 1.0 e - 6

predicts a δΦ residual at LIGO O5 sensitivity

LIGO catalog benchmarks

GW150914 chirp mass

Abbott et al., PRL 116, 061102 (2016)

Predicted (SPT)

28.096M_⊙

Measured

28.600M_⊙

Δ = 1.76%

M_{c} = (m_{1} m_{2})^{3/5} / (m_{1} + m_{2})^{1/5}

GW170817 chirp mass

Abbott et al., PRL 119, 161101 (2017) — binary NS

PASS

Predicted (SPT)

1.185M_⊙

Measured

1.188M_⊙

Δ = 0.27%

M_{c} = (m_{1} m_{2})^{3/5} / (m_{1} + m_{2})^{1/5}

GW190521 chirp mass

Abbott et al., PRL 125, 101102 (2020)

PASS

Predicted (SPT)

65.100M_⊙

Measured

65.600M_⊙

Δ = 0.76%

M_{c} = (m_{1} m_{2})^{3/5} / (m_{1} + m_{2})^{1/5}

GW230529 chirp mass

LIGO/Virgo/KAGRA O4 NS-BH candidate

Predicted (SPT)

1.913M_⊙

Measured

1.940M_⊙

Δ = 1.40%

M_{c} = (m_{1} m_{2})^{3/5} / (m_{1} + m_{2})^{1/5}

ISCO frequency f_ISCO

Schwarzschild geodesic, ISCO at r = 6 GM/c²

PASS

Predicted (SPT)

67.6Hz

Measured

67.7Hz

Δ = 0.0898%

f_{I S C O} = c^{3} / (66 π GM)

Soundness

Masses & distance

m₁ (M☉)36.0 M☉

m₂ (M☉)29.0 M☉

D (Mpc)410 Mpc

ε (SPT correction)10^{-6.0}

LIGO event