Cosmology Dark Energy Equation Of State

The dark energy equation

In cosmology, the equation of state of dark energy is a single number, usually written w, that relates the pressure of dark energy to its energy density. The simplest model, the cosmological constant, has w = -1 exactly. This is the value that fits the observed accelerating expansion of the universe when dark energy is treated as a constant energy density filling space. Any deviation from -1 would mean dark energy behaves differently, either as a dynamic field (quintessence, with w > -1) or as a phantom energy (with w < -1) that could lead to exotic futures.

The standard models are usually grouped into five families: the cosmological constant, quintessence, phantom energy, quintom (which crosses the -1 line), and holographic dark energy. The framework's machine-checked library of formal theorems defines these five models and proves, by direct computation, that there are exactly five of them. The baseline equation of state is set to w = -1, the cosmological constant value, as a definitional starting point.

In Recognition Science, the framework models a discrete record of events, called a ledger, where the cost of recognition is forced by a proved functional equation. That cost function, J(x) = (x + 1/x)/2 - 1, takes a specific value at the golden ratio phi: J(phi) = phi - 3/2, which is approximately 0.118. The framework's cosmology module uses this exact bound as a ceiling. It states that any correction to the dark energy equation of state, written δw, must satisfy |δw| ≤ J(phi), meaning the deviation from w = -1 cannot exceed about 0.118 in magnitude. This places the framework's prediction for the present-day value w_0 in the range (-1 - J(phi), -1), approximately (-1.13, -1).

The bound is derived in closed form, not as a numerical approximation, because the identity 1/phi = phi - 1 simplifies the expression exactly. The module also packages the five-model count and the baseline value into a single certificate structure, with the machine-checked proof that the count is five and that the baseline is -1. The key consequence is that the framework does not simply assume w = -1; it allows a bounded deviation and proves that the deviation is capped by the same cost function that forces the golden ratio elsewhere in the framework.

This matters because it gives a falsifiable prediction: if future measurements of dark energy were to find w_0 below about -1.13, the framework's bound would be violated. The bound is a theorem about the framework's internal structure, not a fit to data. The comparison to astronomical observations remains an empirical check, not a derivation.

/-- Dark energy EoS w_0 = -1 (cosmological constant baseline). -/
def wLambda : ℝ := -1

theorem darkEnergyModelCount : Fintype.card DarkEnergyModel = 5 := by decide

/-- BIT correction bound: `|δw| ≤ J(φ)`. This is the exact phantom-Carnot ceiling
`J(φ) = φ − 3/2 ≈ 0.118` in closed form (not an approximation): since `1/φ = φ − 1`,
the earlier obfuscated form `1/φ − 3/2 + 1` equals `φ − 3/2` exactly. -/
noncomputable def bitCorrectionBound : ℝ := phi - 3 / 2

What this page does not claim

The framework does not prove that dark energy actually deviates from w = -1; it only bounds the possible deviation. The framework does not derive the measured value of the dark energy density or the cosmological constant itself. The framework's bound is not a fit to observational data; it is an internal structural theorem.

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

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