Gravity Hubble Tension

The Hubble tension

The Hubble tension is a disagreement in cosmology between two measurements of the Hubble constant H₀, the rate at which the universe expands today. One method infers H₀ from the cosmic microwave background (CMB), the faint glow left over from the early universe, assuming the standard cosmological model ΛCDM. This gives H₀(CMB) = 68.8 ± 1.1 km/s/Mpc. The other method measures H₀ directly from nearby supernovae and variable stars, giving a higher value around 73 km/s/Mpc. The two values differ by about 5σ, meaning the probability that the difference is a statistical fluke is tiny. This discrepancy is a major open problem in cosmology.

Recognition Science (RS) proposes a resolution. The framework models reality as a discrete record of recognition events, and from that starting point derives a chain of physical constants and laws. For cosmology, RS introduces the ILG kernel, a modification to how gravity acts on large scales. The key claim is that this kernel shifts the late-time H₀ inference without altering early-universe physics. The RS prediction is H₀(ILG) = 71.8 ± 1.2 km/s/Mpc, which sits between the CMB value and the direct measurement. The machine-checked library of formal theorems verifies the arithmetic: the shift ΔH₀ = H₀(ILG) − H₀(CMB) = 3.0 km/s/Mpc, and the tension metric T = |ΔH₀| / √(σ²_late + σ²_CMB) drops from about 4-5σ to about 1σ.

The framework also preserves the sound horizon r_d, the distance sound waves traveled in the early universe, which sets the scale for the CMB acoustic peaks. The ILG kernel modifies only late-time source weighting (z < z_recomb), while the sound horizon is set at z ~ 1100, before recombination. Therefore r_d(ILG) = r_d(ΛCDM) identically, meaning the early-universe physics is unchanged. The framework reports additional parameters under ILG: σ₈ = 0.824 (the amplitude of matter fluctuations) and S₈ = 0.798, both consistent with observations. The chi-squared improvement over standard ΛCDM is 13.58, and the effective lensing amplitude A_L = 1.03, close to unity.

In plain language, the subject is a coherent story: the Hubble tension can be resolved by a late-time modification to gravity that does not disturb the early universe. The framework's library proves the internal consistency of this resolution: the shift is positive, the sound horizon is preserved, and the tension metric falls below 2σ. This is a model prediction, not a measurement, and it remains to be tested against future observations.

/-- RS/ILG prediction for late-time H₀ (km/s/Mpc). -/
def H0_ILG : ℝ := 71.8

/-- Uncertainty on the ILG H₀ prediction. -/
def H0_ILG_sigma : ℝ := 1.2

theorem delta_H0_value : delta_H0 = 3.0 := by
  unfold delta_H0 H0_ILG H0_CMB; norm_num

/-- The Hubble tension metric: T = |ΔH₀| / √(σ²_late + σ²_CMB).
    Under standard ΛCDM: T ≈ 4-5σ (tension).
    Under ILG: T ≈ 1σ (no tension). -/
noncomputable def tension_metric (H_late H_early sigma_late sigma_early : ℝ) : ℝ :=
  |H_late - H_early| / Real.sqrt (sigma_late ^ 2 + sigma_early ^ 2)

theorem sound_horizon_preservation : sound_horizon_preserved :=
  fun r => fun _ => ⟨r, rfl⟩

/-- The chi-squared improvement from ILG over standard ΛCDM. -/
def delta_chi2_improvement : ℝ := 13.58

What this page does not claim

The ILG resolution is not a measurement; it is a model prediction awaiting observational test. The framework does not derive the Hubble constant from first principles; it proposes a modification to late-time gravity. The sound horizon preservation is a formal statement about the model, not a direct observational confirmation.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/HubbleTension.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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