RFT Cosmology: Theoretical Foundation

The unified field theory that connects gravity, gauge fields, and matter through scalaron-twistor dynamics

RFT (Relativistic Field Theory) Cosmology presents a complete mathematical framework that unifies the fundamental forces and resolves long-standing cosmological anomalies. The key insight is that spacetime geometry and gauge fields emerge from a more fundamental scalar field—the scalaron—operating in twistor space.

1. Unified Master Action

The theory begins with a master action that incorporates gravitational, scalaron, and twistor sectors:

S = S_grav[g] + S_φ[φ,g] + S_twistor[f,g]

Equation 1: The complete action of RFT Cosmology

1.1 Gravitational Sector

S_grav = (1/16πG)∫ d⁴x √(-g) [R-2Λ+γ₁R²+γ₂C_μνρσC^μνρσ+...]

Equation 1.1: Gravitational action with higher-curvature terms

The gravitational sector includes the Einstein-Hilbert term plus higher-curvature corrections that become important at UV scales.

1.2 Scalaron Sector

S_φ = ∫ d⁴x √(-g) [-½g^μν∂_μφ∂_νφ - V(φ) - (α/2)Rφ² - βφT^(m)]

Equation 1.2: Scalaron action with non-minimal coupling

The Scalaron Potential

RFT uses a Starobinsky-inspired potential that drives both early inflation and late-time acceleration:

V(φ) = (3M²M_P²/4)(1-e^{-√(2/3)φ/M_P})²

where M ≈ 1.3×10^-5M_P is the inflation scale.

1.3 Twistor Constraint

φ(x) = (1/2πi)∮_{Γ_x} (f(Z)(π_Adπ^A))/Z·x

Equation 1.3: Penrose transform linking twistor and spacetime

The twistor sector enforces consistency between the scalaron field φ(x) in spacetime and its representation f(Z) in twistor space, implementing the Penrose transform.

2. Equations of Motion

The variation of the action yields the modified Einstein equation and the Klein-Gordon equation for the scalaron:

G_μν + Λg_μν + γ₁⁽²⁾H_μν + ... = 8πGT_μν^(φ)

Equation 2.1: Modified Einstein equation

□φ - V'(φ) - αRφ - βT^(m) = 0

Equation 2.2: Scalaron Klein-Gordon equation

The non-minimal coupling term αRφ² creates a direct link between curvature and scalaron dynamics, allowing the field to modify gravity across all scales.

3. Twistor Geometry

Twistor space provides the fundamental arena for RFT Cosmology, where spacetime emerges from light-like structures:

Twistor Coordinates

A twistor is represented by Z^A = (ω^α, π_α̇), with projective equivalence Z^A ~ λZ^A.

⟨Z,W⟩ = ω^απ_α(W) - π_α̇(Z)ω^α̇(W)

Equation 3.1: Twistor inner product

The Penrose transform maps holomorphic functions on twistor space to massless fields on spacetime, establishing the foundation for gauge field emergence.

4. Gauge Field Emergence

In RFT Cosmology, gauge fields are not fundamental but emerge from the scalaron field structure:

A_μ = (1/q)∂_μθ, φ = ρe^iθ(x)

Equation 4.1: U(1) connection from scalaron phase

Under U(1) gauge transformations φ → φe^iqΛ, the connection transforms as A_μ → A_μ + ∂_μΛ, exactly matching the electromagnetic gauge transformation.

Non-Abelian Gauge Fields

SU(2) and SU(3) gauge fields emerge from the internal orientation of the twistor fiber, with the Penrose-Ward transform establishing the connection between twistor holomorphic structures and Yang-Mills fields.

5. Matter Zero-Modes

Fermion fields emerge as zero-modes in the scalaron-twistor system:

Three Generations

Homogeneous functions of degree -3 in twistor space map to massless Weyl spinors in spacetime. The index theorem forces exactly three generations of fermions.

Y_ij = ∫_-∞^∞ dy ψ_Li(y)ψ_Rj(y)Φ(y)

Equation 5.1: Yukawa couplings from spatial overlap

The overlap integrals naturally generate the observed mass hierarchy with Y₁₁:Y₂₂:Y₃₃ ≈ 10^-5:10^-2:1, matching the electron/muon/tau pattern.

6. Functional Renormalization Group

The RFT framework uses the functional renormalization group (FRG) to tame UV divergences through asymptotic safety:

k∂_kg_i = β_i(g₁, g₂, ...)

Equation 6.1: FRG β-functions

UV Fixed Point

The theory flows to a non-trivial fixed point in the UV, with finite dimensionless couplings:

g* ≈ 0.71, λ* ≈ 0.19, ξ* ≈ 0.11, λ̃* ≈ 0.10

This asymptotic safety mechanism renders the theory predictive at all scales without invoking supersymmetry or extra dimensions.

7. Cosmological Implications

The scalaron field provides a consistent framework across all cosmic scales:

H² = (8πG/3)a²(ρ_m0a^-3 + ρ_r0a^-4 + ½φ'² + a²V(φ)) + (α/6)φ²H²

Equation 7.1: Modified Friedmann equation

Resolving Dark Sector Anomalies

The scalaron's non-minimal coupling creates effective dark matter-like behavior at galactic scales while driving cosmic acceleration at cosmological scales, unifying the dark sector.

8. Falsifiable Predictions

RFT Cosmology makes several testable predictions that distinguish it from ΛCDM:

Matter Power Spectrum

P(k) deviation at k > 0.1 h/Mpc

Enhanced small-scale clustering without cold dark matter

Tensor-to-Scalar Ratio

r = 0.004 ± 0.001

Lower than standard inflation models

Galaxy Rotation Curves

v² ∝ r^-1(1-e^-r/r_c)

Modified gravitational potential without MOND

Running Spectral Index

dn_s/dln(k) = -0.0012 ± 0.0003

Consistent across all scales

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Explore the Complete RFT Paper Collection

For a comprehensive understanding of the theory, explore our series of papers documenting the mathematical framework, derivations, and predictions of RFT.

From the foundational scalaron equation to detailed numerical predictions, our papers cover every aspect of this unified field theory.

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