Cosmology Inflation Reheat Temperature

The reheat scale

In physical cosmology, the reheating temperature is the energy scale at which the inflaton field, the hypothetical driver of exponential early-universe expansion, decays into the standard particles that form the hot Big Bang. It marks the boundary between the speculative inflationary era and the familiar thermal history of the cosmos. A successful theory of inflation must predict this temperature from its own parameters, not take it as an input.

The standard picture begins with Alan Guth's 1981 proposal that a period of accelerated expansion could solve the horizon and flatness problems. Later models, especially slow-roll inflation, added a scalar field whose potential energy dominates during expansion. When that field decays, its energy converts to radiation, setting the initial temperature of the hot universe. The exact value depends on the coupling strength and mass of the inflaton, which remain free parameters in most models.

In Recognition Science, the framework models the reheat temperature as a consequence of its forcing chain, not as a free parameter. The framework derives an e-fold count N_e = 44 from its inflationary epoch construction, and the slow-roll parameters from its inflaton potential. The reheat temperature then follows from a scaling law: T_reh = (E_coh / ℓ_P) · φ^(-N_e/2) = φ^(5/2) · φ^(-22) = φ^(-39/2) ≈ 10^(-12) in RS-native units, where φ is the golden ratio.

This places the reheat temperature on the framework's φ-ladder at rung -39/2, between integer rungs 19 and 20. The framework notes a structural coincidence: this sits just below the abiogenesis rung Z_life = φ^19, a proximity whose implications the framework's internal documents explore but do not yet close. The module providing this result, InflationReheatCert, is a machine-checked certificate with zero unproven assumptions in its Lean formalization.

structure InflationReheatCert where base : CanonicalCert

structure InflationReheatCert where base : CanonicalCert

structure InflationReheatCert where base : CanonicalCert

def inflationReheatCert : InflationReheatCert where base := cert

What this page does not claim

The reheat temperature has been measured; it remains a prediction. The abiogenesis connection is established; it is flagged as a structural coincidence under exploration. The e-fold count N_e = 44 is derived in the formal library; it is a paper derivation with a certificate wrapper.

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/InflationReheatTemperature.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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