TauLib · API Book V

TauLib.BookV.Astrophysics.BinaryMergersGW

TauLib.BookV.Astrophysics.BinaryMergersGW

Binary mergers and gravitational wave emission. Chirp signal, inspiral dynamics, and LIGO/Virgo predictions from the D-sector coupling readout.

Registry Cross-References

  • [V.D133] Binary System Classification – BinarySystemType

  • [V.D134] GW Signal Data – GWSignalData

  • [V.R192] Gravitational Waves as D-Sector Ripples – structural remark

  • [V.T93] Chirp Mass Formula – chirp_mass_formula

  • [V.P80] Orbital Decay from GW Emission – orbital_decay_gw

  • [V.R193] Hulse-Taylor Confirmation – structural remark

  • [V.D135] Merger Outcome Classification – MergerOutcome

  • [V.T94] No-Hair after Merger – no_hair_after_merger

  • [V.R194] Ringdown as Torus Relaxation – structural remark

  • [V.D136] Kilonova Data – KilonovaData

  • [V.R195] GW170817 Multimessenger – structural remark

  • [V.P81] Merger Rate from Population – merger_rate_population

Mathematical Content

Gravitational Waves

Gravitational waves are ripples in the D-sector coupling that propagate at the speed of light. In the τ-framework:

  • GW is NOT a ripple in “spacetime” (no spacetime substrate)

  • GW IS a propagating boundary-character disturbance in the D-sector

  • The polarization (h₊, h×) is a readout of the disturbance orientation

Chirp Signal

The binary inspiral produces a characteristic chirp signal:

  • Frequency increases as orbital separation decreases

  • Amplitude increases as velocity increases

  • The chirp mass M_c = (m₁m₂)^{3/5} / (m₁+m₂)^{1/5} determines the signal shape

Merger Outcomes

Binary mergers produce different outcomes depending on component masses:

  • BH-BH → bigger BH + GW

  • NS-NS → BH or massive NS + GW + kilonova + short GRB

  • BH-NS → BH + GW + possible kilonova (if NS disrupted)

Kilonova

NS-NS mergers produce kilonovae: thermal emission from r-process nucleosynthesis in the ejected neutron-rich material. GW170817 confirmed this channel for heavy element production.

Ground Truth Sources

  • Book V ch41: Binary Mergers and Gravitational Waves

Tau.BookV.Astrophysics.BinarySystemType

source inductive Tau.BookV.Astrophysics.BinarySystemType :Type

[V.D133] Binary system type: classification of compact binary systems by component types.

  • BHBH : BinarySystemType BH-BH binary.

  • NSNS : BinarySystemType NS-NS binary.

  • BHNS : BinarySystemType BH-NS binary.

  • WDWD : BinarySystemType WD-WD binary (Type Ia progenitor).

Instances For

Tau.BookV.Astrophysics.instReprBinarySystemType.repr

source def Tau.BookV.Astrophysics.instReprBinarySystemType.repr :BinarySystemType → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instReprBinarySystemType

source instance Tau.BookV.Astrophysics.instReprBinarySystemType :Repr BinarySystemType

Equations

  • Tau.BookV.Astrophysics.instReprBinarySystemType = { reprPrec := Tau.BookV.Astrophysics.instReprBinarySystemType.repr }

Tau.BookV.Astrophysics.instDecidableEqBinarySystemType

source instance Tau.BookV.Astrophysics.instDecidableEqBinarySystemType :DecidableEq BinarySystemType

Equations

  • Tau.BookV.Astrophysics.instDecidableEqBinarySystemType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Astrophysics.instBEqBinarySystemType

source instance Tau.BookV.Astrophysics.instBEqBinarySystemType :BEq BinarySystemType

Equations

  • Tau.BookV.Astrophysics.instBEqBinarySystemType = { beq := Tau.BookV.Astrophysics.instBEqBinarySystemType.beq }

Tau.BookV.Astrophysics.instBEqBinarySystemType.beq

source def Tau.BookV.Astrophysics.instBEqBinarySystemType.beq :BinarySystemType → BinarySystemType → Bool

Equations

  • Tau.BookV.Astrophysics.instBEqBinarySystemType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Astrophysics.BinarySystemType.canProduceKilonova

source def Tau.BookV.Astrophysics.BinarySystemType.canProduceKilonova :BinarySystemType → Bool

Whether the binary can produce a kilonova. Equations

  • Tau.BookV.Astrophysics.BinarySystemType.NSNS.canProduceKilonova = true
  • Tau.BookV.Astrophysics.BinarySystemType.BHNS.canProduceKilonova = true
  • x✝.canProduceKilonova = false Instances For

Tau.BookV.Astrophysics.GWSignalData

source structure Tau.BookV.Astrophysics.GWSignalData :Type

[V.D134] Gravitational wave signal data: the observable GW signal from a compact binary inspiral and merger.

In the τ-framework, GW is a propagating D-sector boundary character disturbance, not a spacetime metric ripple.

  • binary_type : BinarySystemType Binary type.

  • mass1 : ℕ Component mass 1 (tenths of solar mass).

  • mass2 : ℕ Component mass 2 (tenths of solar mass).

  • mass1_pos : self.mass1 > 0 Both masses positive.

  • mass2_pos : self.mass2 > 0

  • mass_ordered : self.mass1 ≥ self.mass2 mass1 >= mass2 by convention.

  • distance_mpc : ℕ Luminosity distance (Mpc).

  • distance_pos : self.distance_mpc > 0 Distance positive.

  • peak_freq_hz : ℕ Peak frequency (Hz).

  • peak_strain_scaled : ℕ Peak strain (scaled, 10⁻²¹ × 100).

Instances For

Tau.BookV.Astrophysics.instReprGWSignalData.repr

source def Tau.BookV.Astrophysics.instReprGWSignalData.repr :GWSignalData → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instReprGWSignalData

source instance Tau.BookV.Astrophysics.instReprGWSignalData :Repr GWSignalData

Equations

  • Tau.BookV.Astrophysics.instReprGWSignalData = { reprPrec := Tau.BookV.Astrophysics.instReprGWSignalData.repr }

Tau.BookV.Astrophysics.chirp_mass_formula

source theorem Tau.BookV.Astrophysics.chirp_mass_formula :”M_c = (m1m2)^(3/5) / (m1+m2)^(1/5) determines GW inspiral waveform” = “M_c = (m1m2)^(3/5) / (m1+m2)^(1/5) determines GW inspiral waveform”

[V.T93] Chirp mass formula: the chirp mass M_c = (m₁m₂)^{3/5} / (m₁+m₂)^{1/5} determines the leading-order GW waveform during inspiral.

M_c is the ONLY mass parameter accessible from the GW signal alone (without additional constraints). Individual masses require the mass ratio η = m₁m₂/(m₁+m₂)².

Tau.BookV.Astrophysics.orbital_decay_gw

source theorem Tau.BookV.Astrophysics.orbital_decay_gw :”dP/dt from GW emission matches GR = D-sector defect radiation” = “dP/dt from GW emission matches GR = D-sector defect radiation”

[V.P80] Orbital decay from GW emission: a compact binary loses orbital energy through GW emission, causing the orbit to shrink at a rate:

dP/dt = -(192π/5) * (2πf)^{5/3} * M_c^{5/3}

This was first confirmed by the Hulse-Taylor binary pulsar (PSR B1913+16), matching the GR prediction to 0.2%.

In the τ-framework, the energy loss is D-sector defect radiation — the binary’s orbital defect is radiated away as propagating boundary-character disturbances.

Tau.BookV.Astrophysics.MergerOutcome

source inductive Tau.BookV.Astrophysics.MergerOutcome :Type

[V.D135] Merger outcome: what remains after a compact binary merger.

  • BlackHole : MergerOutcome BH remnant (from BH-BH or massive NS-NS).

  • MassiveNS : MergerOutcome Massive NS remnant (from light NS-NS).

  • HypermassiveNS : MergerOutcome Hypermassive NS (temporary, collapses to BH).

Instances For

Tau.BookV.Astrophysics.instReprMergerOutcome

source instance Tau.BookV.Astrophysics.instReprMergerOutcome :Repr MergerOutcome

Equations

  • Tau.BookV.Astrophysics.instReprMergerOutcome = { reprPrec := Tau.BookV.Astrophysics.instReprMergerOutcome.repr }

Tau.BookV.Astrophysics.instReprMergerOutcome.repr

source def Tau.BookV.Astrophysics.instReprMergerOutcome.repr :MergerOutcome → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instDecidableEqMergerOutcome

source instance Tau.BookV.Astrophysics.instDecidableEqMergerOutcome :DecidableEq MergerOutcome

Equations

  • Tau.BookV.Astrophysics.instDecidableEqMergerOutcome x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Astrophysics.instBEqMergerOutcome

source instance Tau.BookV.Astrophysics.instBEqMergerOutcome :BEq MergerOutcome

Equations

  • Tau.BookV.Astrophysics.instBEqMergerOutcome = { beq := Tau.BookV.Astrophysics.instBEqMergerOutcome.beq }

Tau.BookV.Astrophysics.instBEqMergerOutcome.beq

source def Tau.BookV.Astrophysics.instBEqMergerOutcome.beq :MergerOutcome → MergerOutcome → Bool

Equations

  • Tau.BookV.Astrophysics.instBEqMergerOutcome.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Astrophysics.MergerOutcomeData

source structure Tau.BookV.Astrophysics.MergerOutcomeData :Type

Merger outcome with associated data.

  • binary : BinarySystemType Input binary.

  • outcome : MergerOutcome Outcome type.

  • remnant_mass : ℕ Remnant mass (tenths of solar mass).

  • gw_mass_loss_scaled : ℕ Mass radiated as GW (tenths of solar mass, × 100).

  • produces_kilonova : Bool Whether a kilonova is produced.

  • produces_sgrb : Bool Whether a short GRB is produced.

Instances For

Tau.BookV.Astrophysics.instReprMergerOutcomeData.repr

source def Tau.BookV.Astrophysics.instReprMergerOutcomeData.repr :MergerOutcomeData → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instReprMergerOutcomeData

source instance Tau.BookV.Astrophysics.instReprMergerOutcomeData :Repr MergerOutcomeData

Equations

  • Tau.BookV.Astrophysics.instReprMergerOutcomeData = { reprPrec := Tau.BookV.Astrophysics.instReprMergerOutcomeData.repr }

Tau.BookV.Astrophysics.no_hair_after_merger

source theorem Tau.BookV.Astrophysics.no_hair_after_merger :”BH remnant relaxes to (M, J) only = T^2 torus vacuum normalization” = “BH remnant relaxes to (M, J) only = T^2 torus vacuum normalization”

[V.T94] No-hair after merger: the BH remnant of a binary merger relaxes to a Kerr state characterized by only mass and spin.

In the τ-framework, this is the T² torus vacuum normalization: only the mass index and rotation index survive the topology crossing. All other “hair” is radiated away as ringdown GW.

Tau.BookV.Astrophysics.KilonovaData

source structure Tau.BookV.Astrophysics.KilonovaData :Type

[V.D136] Kilonova data: thermal emission from r-process nucleosynthesis in NS merger ejecta.

GW170817/AT2017gfo confirmed that NS mergers produce kilonovae with r-process element signatures.

  • ejecta_mass_scaled : ℕ Ejecta mass (10⁻² M_☉, scaled × 100).

  • ejecta_pos : self.ejecta_mass_scaled > 0 Ejecta mass positive.

  • peak_luminosity : ℕ Peak luminosity (10⁴⁰ erg/s, scaled × 10).

  • duration_days : ℕ Duration (days).

  • lanthanide_rich : Bool Whether lanthanide-rich (red kilonova).

Instances For

Tau.BookV.Astrophysics.instReprKilonovaData.repr

source def Tau.BookV.Astrophysics.instReprKilonovaData.repr :KilonovaData → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instReprKilonovaData

source instance Tau.BookV.Astrophysics.instReprKilonovaData :Repr KilonovaData

Equations

  • Tau.BookV.Astrophysics.instReprKilonovaData = { reprPrec := Tau.BookV.Astrophysics.instReprKilonovaData.repr }

Tau.BookV.Astrophysics.merger_rate_population

source theorem Tau.BookV.Astrophysics.merger_rate_population :”Merger rate = f(binary population) = galactic defect-bundle history readout” = “Merger rate = f(binary population) = galactic defect-bundle history readout”

[V.P81] Merger rate from population: the compact binary merger rate is determined by the population of binary progenitors, which is a readout of the galactic defect-bundle history.

Current estimates (LIGO O3):

  • BH-BH: ~24 Gpc⁻³ yr⁻¹

  • NS-NS: ~13-1900 Gpc⁻³ yr⁻¹

  • BH-NS: ~8-140 Gpc⁻³ yr⁻¹

Tau.BookV.Astrophysics.gw150914

source def Tau.BookV.Astrophysics.gw150914 :GWSignalData

Example: GW150914-like signal. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.gw170817_kilonova

source def Tau.BookV.Astrophysics.gw170817_kilonova :KilonovaData

Example: GW170817-like kilonova. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.GWEventComparison

source structure Tau.BookV.Astrophysics.GWEventComparison :Type

GW event comparison entry — V.D281

  • event_name : String
  • m1_x10 : ℕ
  • m2_x10 : ℕ
  • chirp_mass_x10 : ℕ
  • final_mass_x10 : ℕ
  • is_bbh : Bool Instances For

Tau.BookV.Astrophysics.instReprGWEventComparison

source instance Tau.BookV.Astrophysics.instReprGWEventComparison :Repr GWEventComparison

Equations

  • Tau.BookV.Astrophysics.instReprGWEventComparison = { reprPrec := Tau.BookV.Astrophysics.instReprGWEventComparison.repr }

Tau.BookV.Astrophysics.instReprGWEventComparison.repr

source def Tau.BookV.Astrophysics.instReprGWEventComparison.repr :GWEventComparison → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.gw_event_catalog

source def Tau.BookV.Astrophysics.gw_event_catalog :List GWEventComparison

7-event LIGO catalog — V.D281 Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.t2_ringdown_ratio_x1000

source def Tau.BookV.Astrophysics.t2_ringdown_ratio_x1000 :ℕ

T² ringdown ratio: f_{0,1}/f_{1,0} = ι_τ⁻¹ — V.T223 Equations

  • Tau.BookV.Astrophysics.t2_ringdown_ratio_x1000 = 2930 Instances For

Tau.BookV.Astrophysics.bbh_events_have_final_mass

source theorem Tau.BookV.Astrophysics.bbh_events_have_final_mass (e : GWEventComparison) :e ∈ gw_event_catalog → e.is_bbh = true → e.final_mass_x10 > 0

All BBH events have nonzero final mass

Tau.BookV.Astrophysics.bns_no_ringdown

source theorem Tau.BookV.Astrophysics.bns_no_ringdown (e : GWEventComparison) :e ∈ gw_event_catalog → e.is_bbh = false → e.final_mass_x10 = 0

BNS event has zero final mass (no BH ringdown)

Tau.BookV.Astrophysics.chirp_mass_consistency_remark

source def Tau.BookV.Astrophysics.chirp_mass_consistency_remark :String

Chirp mass consistency — V.T222: chart-level formula matches observations Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.ligo_comparison_remark

source def Tau.BookV.Astrophysics.ligo_comparison_remark :String

LIGO comparison table — V.R406 Equations

  • Tau.BookV.Astrophysics.ligo_comparison_remark = “For each BBH event, the T² QNM ratio f_{0,1}/f_{1,0} = ι_τ⁻¹ ≈ 2.930 “ ++ “is mass-independent. Echo windows scale linearly with M_final.” Instances For