Cosmology Cosmological Constant Dark Energy Scale E V

The dark energy scale

The cosmological constant Λ appears in Einstein's field equations as a term that pushes space apart. Its measured value is about 1.1 × 10⁻⁵² inverse square meters, which corresponds to an energy density of roughly 6 × 10⁻²⁷ kilograms per cubic meter. In particle physics units, that density translates to an energy scale near 0.002 electronvolts, or 2 × 10⁻³ eV. This is the number the declaration dark_energy_scale_eV records: a definitional choice in the framework's library, not a derived result.

The problem this scale exposes is stark. Quantum field theory, the standard framework for particle physics, estimates the vacuum energy density from known fields and gets a value about 10¹²⁰ times larger than what astronomers observe. This mismatch, the worst fine-tuning problem in physics, is what the framework's cosmological_constant_problem theorem names. The theorem itself is a placeholder: it states the problem exists, but it does not resolve it.

In Recognition Science, the framework models the vacuum not as empty space but as carrying a baseline cost from its ledger, a discrete record of recognition events. The framework's library contains a definition, vacuumJCost, that sets this baseline cost to the J-cost function evaluated at the golden ratio φ. A companion theorem, jcost_cancellation, asserts that most of the vacuum energy cancels, leaving a tiny residual that the framework identifies as the cosmological constant. That theorem is also a placeholder: it states the cancellation as a proposition without providing the mechanism.

The framework's library also defines the equation of state for dark energy as exactly −1, matching the simplest form of a cosmological constant, and lists observational status values: Λ = (1.1 ± 0.01) × 10⁻⁵² m⁻² and w = −1.03 ± 0.03. These are recorded definitions, not measurements made by the framework. The library's falsifier structure names the conditions that would kill the framework's approach: if w is not −1, or if Λ varies over time, the mechanism fails.

What the declaration does not do is derive the 2 × 10⁻³ eV scale from first principles. It is a fixed input, a number chosen to match the observed value, not a consequence of the forcing chain that produces other constants in the framework. The framework's own notes list the phi-ladder hypothesis, which suggests φ-scaling might explain why Λ is small but nonzero, but that hypothesis is not proved. The cosmological constant problem remains open in the framework, exactly as it remains open in conventional physics.

/-- The observed cosmological constant Λ ≈ 1.1 × 10⁻⁵² m⁻². -/
noncomputable def lambda_observed : ℝ := 1.1e-52

/-- The dark energy scale in eV: (ρ_Λ c² / ℏ³ c³)^(1/4) ≈ 2 meV. -/
noncomputable def dark_energy_scale_eV : ℝ := 2e-3  -- eV

/-- Naive QFT prediction: ρ_vac ~ m_P⁴ / (ℏ³ c³) ~ 10⁹⁶ kg/m³.

    This is 10¹²³ times larger than observed!

    Even with supersymmetry cutoff at 1 TeV:
    ρ_SUSY ~ (1 TeV)⁴ ~ 10⁴⁸ kg/m³

    Still 10⁷⁵ times too large! -/
theorem cosmological_constant_problem :
    -- ρ_predicted / ρ_observed ~ 10¹²³
    -- This is the most extreme fine-tuning in physics
    True := trivial

/-- Key insight: In RS, the cosmological constant arises from
    the DIFFERENCE between positive and negative J-cost contributions.

    1. Positive contributions: Each field mode adds ~E_P
    2. Negative contributions: φ-structure provides cancellation
    3. Residual: The tiny observed Λ

    Λ_eff = Λ_bare - Λ_φ-cancel + Λ_residual

    The residual is ~10⁻¹²² of the bare value! -/
theorem jcost_cancellation :
    -- Most of the vacuum energy cancels
    -- Only a tiny residual remains
    -- This residual IS the cosmological constant
    True := trivial

/-- Dark energy equation of state: w = p/ρ = -1.

    For a cosmological constant, pressure equals negative density.
    This drives accelerated expansion. -/
noncomputable def equationOfState : ℝ := -1

What this page does not claim

The 2 × 10⁻³ eV scale is not derived from the framework's forcing chain; it is a fixed input matched to observation. The jcost_cancellation theorem does not provide a mechanism for the cancellation; it only states that cancellation occurs. The framework does not claim to resolve the cosmological constant problem; its own notes call the resolution a target.

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/CosmologicalConstant.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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