Cosmology Gravitational Wave Background3

The background's ledger

A stochastic gravitational wave background is the random, persistent wash of gravitational waves from many independent sources across cosmic history, like the hiss of static on a radio tuned between stations. Astronomers measure it as an energy density, Omega_GW, the fraction of the universe's critical energy carried by these waves. Pulsar timing arrays such as NANOGrav and PPTA probe this background at nanohertz frequencies, where current measurements place Omega_GW around 10^-9, a signal that has grown from a detection candidate into a steadily mapped feature of the cosmos.

The standard picture of this background is a sum over astrophysical and cosmological sources: supermassive black hole binaries, cosmic strings, and processes from the early universe. Each source contributes its own spectral shape, and the total is a smooth, nearly scale-invariant hum. The measurement is hard because the signal is faint and the noise from individual pulsars is large, but the statistical correlation across many pulsars is what lifts the background out of the noise.

In Recognition Science, the framework models this background through a ledger, a discrete record of recognition events, where the cost of recognizing a mass m against an energy scale e is forced by a proved cost function. The module defines this cost as Jcost(m/e), the same universal cost function that the framework's core theorem derives from five plain conditions. The module proves three general facts about this cost: it vanishes when m equals e, it is never negative for positive inputs, and the golden ratio phi minus 3/2 is positive. These are facts about the cost function itself, not about gravitational waves.

The research note attached to the module sketches an intended application: if the background's energy density were written as Omega_GW = J(phi)^2 times Omega_matter, the framework's numbers would give 0.014 times 0.315, or 0.0044, far above the measured 10^-9. That comparison is structural, a note about where the idea was meant to go, not a result. The module itself proves nothing specific to the gravitational wave background, because the cost function is defined without reference to any particular mass or energy scale. What would make it a theorem about the background is a definition of m and e in the background's own terms.

What the module does establish is a template. It shows that the framework's cost function, applied to a ratio of two scales, has the basic properties any physical energy density should have: zero at equality, positivity away from it, and a positive threshold tied to the golden ratio. The work of connecting that template to the measured background remains open, and the gap between the sketched 0.0044 and the measured 10^-9 is a clear statement of how far the framework is from a physical prediction here.

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

This module does not predict or derive the measured value of the stochastic gravitational wave background. The framework's cost function is not shown to be the correct energy density for gravitational waves in this module. The research note's comparison of 0.0044 to 10^-9 is not a proved result, only a stated structural mismatch.

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/GravitationalWaveBackground3.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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