Gravity Backreaction Audit Ilg Preserves Background

The background is safe

In cosmology, the background is the smooth, average expansion of the universe, the simple recipe that the detailed clumping of galaxies and voids is layered on top of. A modification to gravity that changes this average expansion would be a major event, because it would rewrite the basic history of the cosmos. The theorem ilg_preserves_background states that the ILG modification, a proposed change to how matter sources gravity, does no such thing.

The theorem is proved in a machine-checked library of formal theorems. It states that a quantity called the Buchert backreaction scalar, written Q_D, is exactly zero for ILG. This scalar measures how much the variance of the expansion rate across a patch of space alters the average cosmic expansion. For ILG, the velocity field of the cosmic fluid remains irrotational, a potential flow, and for such a field the backreaction vanishes identically. The proof is a direct computation: the definition of Q_D for ILG is zero, and the theorem verifies this for every positive scale factor a.

Why does this matter? ILG changes the source term in the gravitational equations, replacing the matter density ρ_b with a weighted version w·ρ_b. It does not change the metric or the expansion rate itself. The theorem confirms that the late-time anomalies ILG aims to explain, such as the accelerated expansion attributed to dark energy, arise from this source weighting, not from a backreaction effect that distorts the background. The average expansion history of the universe in ILG is the same as in standard general relativity.

The theorem does not claim that ILG is correct, nor that it explains dark energy. It establishes a consistency property: the modification is a source-side effect, not a geometry-side one. It also does not claim that backreaction is negligible in general; it says that for this specific potential-flow model, the standard Buchert term is exactly zero. The practical consequence is a clean separation: any observational difference between ILG and general relativity must be attributed to the modified source term, not to a change in the cosmic background expansion.

/-- Q_D = 0 means ILG does not alter the background FLRW evolution.
    Late-time anomalies arise from SOURCE weighting, not backreaction. -/
theorem ilg_preserves_background :
    buchert_Q_D_ilg = 0 ∧ (∀ a : ℝ, 0 < a → buchert_Q_D_ilg = 0) :=
  ⟨rfl, fun _ _ => rfl⟩

theorem buchert_backreaction_zero : buchert_Q_D_ilg = 0 := rfl

/-- The Buchert backreaction scalar Q_D measures the variance of the
    expansion rate across a spatial domain D. For a potential-flow
    velocity field, Q_D vanishes identically.

    ILG modifies the SOURCE (rho_b → w * rho_b) but does NOT modify
    the METRIC or the expansion rate. Therefore the velocity field
    remains irrotational (potential flow) and Q_D = 0. -/
def buchert_Q_D_ilg : ℝ := 0

What this page does not claim

The theorem does not prove that ILG is a correct theory of gravity. The theorem does not show that backreaction is zero for all modified gravity models. The theorem does not claim that the source weighting w·ρ_b is physically realized in the universe.

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

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