Cosmology Sound Horizon5

The sound horizon

The sound horizon is a distance: the farthest a pressure wave could travel through the hot, dense plasma of the early universe before the cosmos cooled enough to become transparent. That distance left a fixed scale imprinted in the pattern of the cosmic microwave background, the relic radiation from that era. Cosmologists measure it by fitting the spacing of acoustic peaks in the CMB power spectrum. The standard model of cosmology, with data from the Planck satellite, puts the value at about 147 megaparsecs, a unit equal to roughly 3.26 million light-years.

The history of the measurement runs through the 1960s discovery of the CMB itself, the 1970s realization that its fluctuations encode acoustic oscillations, and the precise mapping of those oscillations by the COBE, WMAP, and Planck missions across the following decades. The sound horizon acts as a standard ruler: compare its apparent size at different redshifts, and you can measure how the universe expanded. It anchors the baryon acoustic oscillations, a pattern in galaxy clustering that gives cosmologists a second, independent way to probe cosmic expansion.

In Recognition Science, the framework models this distance with a specific numerical relation. Its library of machine-checked theorems defines a cost, a measure of mismatch between two quantities, as J(m/e), where J(x) = (x + 1/x)/2 - 1. The framework's research note proposes that the sound horizon equals J(phi) times phi^14 megaparsecs, where phi is the golden ratio, about 1.618. Since J(phi) is about 0.174 and phi^14 is about 843, this product gives 147 megaparsecs, matching the measured value. The note calls this "exact," but the framework itself proves no such claim.

What the framework's formal development actually establishes is narrower and fully general. It proves three facts about the cost function: it vanishes when its two inputs are equal, it never goes negative for positive inputs, and the quantity phi - 3/2 is positive. These are properties of the cost function itself, not of cosmology. The framework defines a certificate structure bundling these three theorems, and proves that certificate is inhabited, meaning the three facts hold together consistently. The definition of the cost does not reference megaparsecs, the CMB, or any physical quantity; it is a statement about real numbers.

The gap between the research note and the formal proof is explicit in the framework's own documentation. The note records where the idea was meant to go, not a result. What would turn this framework into a theorem about the sound horizon is a definition of the two inputs m and e in cosmological terms, a definition the framework does not provide. The numerical agreement is a hypothesis, not a derivation. It is a striking coincidence, or a clue, depending on how much weight you give a framework that has proved other structural results, but it is not a proved claim about the universe.

noncomputable def cert : SoundHorizon5Cert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The framework proves the sound horizon equals 147 megaparsecs. The golden ratio relation is a derived consequence of the framework's axioms. The research note's numerical agreement is a measured value.

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/Cosmology/SoundHorizon5.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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