Cosmology Dark Energy Cosmological Constant

The declaration's meaning

The cosmological constant, usually written Λ, is the leading explanation for dark energy, the unknown agent that makes the expansion of the universe accelerate. In standard physics, Λ is a free parameter in Einstein's field equations, a number that must be measured and inserted by hand. Its measured value is extraordinarily small, about 10⁻¹²² in Planck units, and its origin is one of the deepest unsolved problems in physics.

In Recognition Science, the declaration cosmologicalConstant is a definition, not a derived result. It sets Λ equal to three times the square of the Hubble constant, H₀, in natural units where the speed of light c equals 1. The Hubble constant measures the current expansion rate of the universe, approximately 2.2 × 10⁻¹⁸ per second. The definition is written as Λ = 3 H₀², which is a choice about how to express the constant in terms of the expansion rate, not a proof that this relationship must hold.

The surrounding framework provides the motivation for this choice. It models the universe as a ledger, a discrete record of events, where the total cost of recognition must balance globally. As space expands, new volume creates room for new entries, and the framework proposes that maintaining this balance generates a residual energy density, which it identifies with Λ. This is a hypothesis about the origin of dark energy, not an established result.

What the declaration does establish, through simple definitions, is a set of numerical properties. The result lambda_positive shows that the defined constant is positive, consistent with the observed accelerating expansion. The result dark_energy_dominates shows that the ratio of dark energy density to the critical density, set at 0.68 in the definition omegaLambda, is greater than 0.5, meaning dark energy dominates the universe's energy budget. The result dark_energy_eos shows that the equation of state, the ratio of pressure to energy density, equals exactly -1, matching the value for a cosmological constant. These are consequences of the definitions, not independent physical discoveries.

The framework's library also includes a falsifier structure, a formal way to record observations that could contradict the prediction. It lists the predicted equation of state as -1 and the predicted scaling of Λ with H₀². The declaration itself, however, does not claim to derive the numerical value of Λ from first principles. The value of H₀ is an input, taken from measurement, and the relationship Λ = 3 H₀² is a definitional choice. The framework's account of why dark energy exists remains a hypothesis, and the precise numerical agreement with observation is not established by these declarations.

/-- The tension energy density is the cosmological constant. -/
noncomputable def cosmologicalConstant : ℝ :=
  -- Λ ≈ (energy to maintain ledger balance) / volume
  -- This scales as H₀² due to the expansion rate
  3 * H0^2  -- In natural units with c = 1

/-- **THEOREM**: The cosmological constant is positive (repulsive). -/
theorem lambda_positive : cosmologicalConstant > 0 := by
  unfold cosmologicalConstant H0
  norm_num

/-- **THEOREM**: Dark energy dominates the universe today. -/
theorem dark_energy_dominates : omegaLambda > 0.5 := by
  unfold omegaLambda
  norm_num

/-- **THEOREM**: Dark energy has w = -1 (cosmological constant). -/
theorem dark_energy_eos : equationOfState = -1 := rfl

What this page does not claim

The declaration does not derive the numerical value of the cosmological constant from first principles. The declaration does not prove that dark energy emerges from ledger tension; that remains a hypothesis. The declaration does not establish that Λ scales as H₀² in a way that matches observation.

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/DarkEnergy.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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