Physics Inflationary Cosmology From Rs

The five-model catalog

Inflationary cosmology is the leading theory of the universe's first fraction of a second. It proposes that space expanded exponentially fast, smoothing out irregularities and seeding the large-scale structure we observe today. The theory is not a single equation but a family of models, each with a different mechanism driving that rapid expansion. The most prominent are Starobinsky inflation, natural inflation, Higgs inflation, chaotic inflation, and axion monodromy. Each produces slightly different predictions for the cosmic microwave background, which is how cosmologists test them against data.

Starobinsky inflation, proposed by Alexei Starobinsky in 1980, uses a correction to Einstein's equations. Natural inflation, introduced in 1990, uses a cosine potential, a periodic energy profile. Higgs inflation, from 2008, couples the Higgs boson to gravity with a non-minimal coupling. Chaotic inflation, proposed by Andrei Linde in 1983, uses a simple power-law potential. Axion monodromy, developed around 2010, uses a linear potential from string theory. These five are not an arbitrary list; they cover the main mechanisms cosmologists actually work with.

In Recognition Science, a framework that derives physical structure from a forced cost of recognition, these five models are not a coincidence. The framework's library, a machine-checked collection of formal theorems, defines an inductive type with exactly these five constructors and proves that the count is five. It also sets the number of e-folds, the measure of how much expansion occurred, to 44. This value comes from the framework's internal structure, specifically a quantity called the baryon rung, rather than from observation.

The framework's contribution is a structural claim: the five canonical models are not a random assortment but a complete set. The count of five matches the framework's derived configuration dimension of 5, suggesting that the number of inflation models is forced by deeper structure. The e-fold value of 44 sits within the range cosmologists typically use, which is between 50 and 60, though it is on the lower end. This is a prediction about the duration of inflation, not a measurement.

The practical consequence is a target for observation. If inflation lasted exactly 44 e-folds, the primordial gravitational wave signal would be at a specific level, larger than the simplest models predict. The framework does not derive the full dynamics of inflation; it provides a catalog and a number. The catalog is formally proved; the number is a definitional choice within the framework. What remains open is whether the 44 e-fold prediction survives contact with the next generation of cosmic microwave background experiments.

theorem inflationModelCount : Fintype.card InflationModel = 5 := by decide

/-- E-folds N_e = 44 (baryonRung). -/
def Nefolds : ℕ := 44

What this page does not claim

The framework does not derive the full dynamics of any inflation model. The e-fold value of 44 is not measured but chosen from the framework's internal structure. The framework does not prove that these five models are the only possible ones in conventional physics.

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

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