MODIFIED FRIEDMANN EQUATION

Standard Friedmann Equation

The classical Friedmann equation from General Relativity describes the expansion rate of a homogeneous, isotropic universe:

(ȧ/a)² = (8πG/3)ρ - k/a² + Λ/3

Where:

• ȧ/a = H(t) — Hubble parameter (expansion rate)
• G — Newton's gravitational constant (6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻²)
• ρ — Total energy density (matter + radiation + dark energy)
• k — Spatial curvature parameter (k = -1, 0, +1)
• a(t) — Scale factor (normalized to a₀ = 1 today)
• Λ — Cosmological constant (~1.1 × 10⁻⁵² m⁻²)

Formal Derivation from Variational Principle

Starting from the grace-modified Einstein-Hilbert action:

S = ∫d⁴x √(-g) [(R - 2G(t, Ψ))/(16πG) + L_m + L_χ]

Variation with respect to g_μν yields:

G_μν + G(t, Ψ)·g_μν = 8πG·T_μν + κ·χ_μν

For FLRW metric ds² = -dt² + a²(t)[dr²/(1-kr²) + r²dΩ²]:

G_00 = 3(ȧ/a)² + 3k/a²

Setting G_00 = 8πGρ + G(t, Ψ) yields the modified Friedmann equation.

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