Physics Bao Bao Peak Approximately 150

The 294 Mpc peak

Baryon acoustic oscillations (BAOs) are the frozen ripples in the distribution of matter across the cosmos, imprinted when pressure waves traveled through the hot plasma of the early universe. The standard cosmological picture, built on the Friedmann equations, predicts that these ripples appear as a characteristic bump in the clustering of galaxies, located at a scale set by the distance sound could travel before recombination, the sound horizon. The Recognition Science framework, which derives physical constants from a forced cost function, has encoded this picture into a machine-checked library of formal theorems, and one declaration in that library, bao_peak_approximately_150, pins down the location of the first peak.

The theorem states that the absolute difference between twice the sound horizon and 294 megaparsecs is less than 1 megaparsec. In symbols, |bao_correlation_peak sound_horizon_approx - 294| < 1. Since the correlation peak is defined as twice the sound horizon, and the sound horizon is set to 147 megaparsecs, the arithmetic is direct: 2 times 147 equals 294. The theorem's content is not the discovery of a new physical law; it is the formal verification that the framework's own definitions, when combined, produce a number consistent with the observed BAO scale. The proof is a simple calculation, checked by the Lean kernel, confirming that the framework's internal model does not contradict the standard result.

The declaration sits atop a chain of definitions that build the BAO picture from the framework's cosmological parameters. The sound horizon itself is an abbreviation set to 147 megaparsecs, a value the framework derives from its predicted baryon density, matter density, and spectral index. The framework also defines the sound speed in the baryon-photon fluid, the baryon loading ratio, and the wavenumber of the BAO peaks. The theorem bao_peaks_evenly_spaced shows that these peaks are evenly spaced in wavenumber, a property that follows from the definition of the peak wavenumber as an integer multiple of pi divided by the sound horizon. These are all formal definitions and theorems within the framework's own model, not independent measurements.

What the declaration does not claim is just as important as what it proves. It does not claim that the framework has derived the sound horizon from first principles; the value of 147 megaparsecs is an input, not an output, of this particular theorem. It does not claim that the framework's cosmological parameters are correct, only that they are internally consistent with the BAO peak location. The theorem is a consistency check, not a derivation. The framework's broader claim, that its cost function forces the observed values of baryon density and spectral index, is a separate matter, and the BAO theorem does not provide evidence for it. The declaration is a small but solid piece of the framework's larger edifice, a formal guarantee that one of its predictions lines up with the standard cosmological story.

theorem bao_peak_approximately_150 :
    |bao_correlation_peak sound_horizon_approx - 294| < 1 := by
  norm_num [bao_correlation_peak, sound_horizon_approx]

/-- The correlation function peak in real space: r = 2π/k₁ ≈ 150 h⁻¹ Mpc. -/
noncomputable def bao_correlation_peak (r_s : ℝ) : ℝ := 2 * r_s

/-- **SOUND HORIZON** (simplified formula valid for matter+radiation):
    r_s ≈ (2/3) × (c / k_eq) × √(6/R_eq) × ln[...]
    where k_eq is the wavenumber at matter-radiation equality.

    Numerically: r_s ≈ 147 Mpc. -/
abbrev sound_horizon_approx : ℝ := 147  -- in Mpc

/-- BAO peaks are evenly spaced in k-space. -/
theorem bao_peaks_evenly_spaced (n m : ℕ) (r_s : ℝ) (hr : 0 < r_s) :
    bao_peak_wavenumber (n + m) r_s = bao_peak_wavenumber n r_s + bao_peak_wavenumber m r_s := by
  unfold bao_peak_wavenumber
  push_cast
  ring

/-- BAO peaks in the matter power spectrum occur at k_n = nπ/r_s. -/
noncomputable def bao_peak_wavenumber (n : ℕ) (r_s : ℝ) : ℝ :=
  n * Real.pi / r_s

What this page does not claim

The theorem does not derive the sound horizon value of 147 megaparsecs from the framework's cost function. The theorem does not prove that the framework's cosmological parameters are the correct ones. The theorem does not establish the existence of baryon acoustic oscillations in the real universe.

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/Physics/BAO.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.

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