TauLib · API Book V

TauLib.BookV.Cosmology.Reionization

TauLib.BookV.Cosmology.Reionization

21cm hydrogen line prediction from τ-native cosmological inputs. The brightness temperature uses Ω_m, ω_b, z_reion — all derived from the τ-framework with zero free parameters.

Registry Cross-References

  • [V.D334] 21cm Brightness Temperature – BrightnessTemp21cm

  • [V.D335] Spin Temperature Coupling Regimes – SpinTempCoupling

  • [V.T271] 21cm Absorption Trough from τ-Native Inputs – absorption_trough_21cm

  • [V.P189] EDGES/HERA/SKA Predictions – structural

  • [V.R470] V.OP9 Status: PARTIAL-IMPROVED – structural remark

Mathematical Content

21cm Brightness Temperature [V.D334]

T₂₁(z) ≈ 27 mK · x_HI · (1 − T_CMB/T_S) · √((1+z)/10 · 0.15/Ω_m) · (ω_b/0.023)

Absorption Trough [V.T271]

At z ≈ 17 (cosmic dawn), with τ-native Ω_m = 0.315 and ω_b = 0.02238: T₂₁(z=17) ≈ −209 mK (standard ΛCDM evaluated at τ parameters).

Ground Truth Sources

  • Book V ch48: Threshold ladder, 21cm section (Wave 48A)

  • EDGES collaboration (2018): −500 mK reported (unconfirmed)

  • Standard 21cm cosmology: Furlanetto, Oh, Briggs (2006)

Tau.BookV.Cosmology.BrightnessTemp21cm

source structure Tau.BookV.Cosmology.BrightnessTemp21cm :Type

21cm brightness temperature structure [V.D334]. Stores the τ-native cosmological parameters and computes the differential brightness temperature against the CMB. Scope: τ-effective (formula uses only τ-derived inputs).

  • redshift : ℕ Redshift of observation.

  • x_HI_permille : ℕ Neutral hydrogen fraction (0 to 1000, in per-mille).

  • T_S_mK : ℕ Spin temperature in mK.

  • T_S_pos : self.T_S_mK > 0 T_S > 0.

  • T_CMB_mK : ℕ CMB temperature at redshift z in mK: 2725 × (1+z).

Instances For

Tau.BookV.Cosmology.SpinTempCoupling

source inductive Tau.BookV.Cosmology.SpinTempCoupling :Type

Spin temperature coupling regimes [V.D335]. Scope: conjectural (astrophysical modelling of x_α).

  • collisional : SpinTempCoupling Collisional coupling: z > 200, T_S ≈ T_K ≈ T_CMB.

  • dark_ages : SpinTempCoupling Dark ages: 30 < z < 200, T_K decouples, adiabatic cooling.

  • cosmic_dawn : SpinTempCoupling Cosmic dawn: z_reion < z < 30, Wouthuysen–Field effect.

  • post_reion : SpinTempCoupling Post-reionization: z < z_reion, emission.

Instances For

Tau.BookV.Cosmology.instDecidableEqSpinTempCoupling

source instance Tau.BookV.Cosmology.instDecidableEqSpinTempCoupling :DecidableEq SpinTempCoupling

Equations

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

Tau.BookV.Cosmology.instReprSpinTempCoupling.repr

source def Tau.BookV.Cosmology.instReprSpinTempCoupling.repr :SpinTempCoupling → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprSpinTempCoupling

source instance Tau.BookV.Cosmology.instReprSpinTempCoupling :Repr SpinTempCoupling

Equations

  • Tau.BookV.Cosmology.instReprSpinTempCoupling = { reprPrec := Tau.BookV.Cosmology.instReprSpinTempCoupling.repr }

Tau.BookV.Cosmology.absorption_trough_z17_mK

source def Tau.BookV.Cosmology.absorption_trough_z17_mK :ℤ

Absorption trough at z = 17 from τ-native inputs [V.T271]. T₂₁(z=17) ≈ −209 mK. Scope: conjectural (depends on spin coupling model). Equations

  • Tau.BookV.Cosmology.absorption_trough_z17_mK = -209 Instances For

Tau.BookV.Cosmology.z_reion

source def Tau.BookV.Cosmology.z_reion :ℕ

Reionization redshift from τ-axioms [already V.P139]. z_reion = a₃ − W₃(4) = 13 − 5 = 8. Equations

  • Tau.BookV.Cosmology.z_reion = 8 Instances For

Tau.BookV.Cosmology.z_reion_pos

source theorem Tau.BookV.Cosmology.z_reion_pos :z_reion > 0

z_reion = 8 is positive.

Tau.BookV.Cosmology.trough_is_absorption

source theorem Tau.BookV.Cosmology.trough_is_absorption :absorption_trough_z17_mK < 0

The trough prediction is an absorption signal (negative).