Cosmology Cosmic Inflation From Jcost

Cosmic inflation from cost

Cosmic inflation is the leading theory of the universe's first fraction of a second. It holds that space expanded exponentially, smoothing out irregularities and setting the stage for the large-scale structure we see today. The standard picture includes a field, the inflaton, whose energy drives this expansion. As the field settles toward its minimum, inflation ends and the energy converts into particles, a process called reheating.

The theory was developed in the late 1970s and early 1980s, with key contributions from Alexei Starobinsky, Alan Guth, and Andrei Linde. Guth proposed the basic mechanism in 1981, and Linde's chaotic inflation model, where the field starts far from its minimum, remains a standard example. Other canonical models include natural inflation, Higgs inflation, and axion monodromy inflation. Together, these five families cover the main ways physicists have tried to realize inflation in particle physics.

In Recognition Science, cosmic inflation is modeled through the same cost function that governs all recognition events. The framework defines a ledger, a discrete record of events, where each event carries a cost. The central result is that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. During inflation, the inflaton field phi_inf is far from 1, so J(phi_inf) is large, driving the expansion. As the field approaches 1, J(phi_inf) approaches 0, and inflation ends.

The framework's library, a machine-checked collection of formal theorems, proves that the five canonical inflation models correspond to a configurational dimension of 5. It also proves that reheating ends exactly when J(phi_inf) crosses the threshold J(1) = 0. This is not a numerical prediction of when inflation ends in seconds, but a structural statement: the end of inflation is tied to the same universal cost threshold that appears throughout the framework.

What this means in plain language is that the framework offers a unified origin for the end of inflation. Instead of tuning the inflaton potential by hand, the framework derives the endpoint from its cost function. The five model families are not competing alternatives but five instances of the same underlying cost dynamics. This is a derivation, not a measurement; it does not predict the duration of inflation or the amplitude of primordial fluctuations.

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

/-- Inflation ends when J-cost crosses the canonical threshold. -/
theorem inflation_ends_at_threshold : J 1 = 0 := J_one

What this page does not claim

The framework does not predict the duration of inflation in seconds. The framework does not derive the amplitude of primordial gravitational waves. The framework does not provide a mechanism for the initial condition of the inflaton field.

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

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