📚 Mathematical Prerequisites

This page uses advanced physics. For full understanding you'll need:

  • General Relativity (Einstein field equations)
  • Quantum Field Theory (gauge theories)
  • Basic differential geometry

👉 Start with our easier explanation

🔬 Framework Status

Mathematical consistency verified
Calculations reproduce known physics
Key predictions await testing (2025-2030)
🤝 Open for peer review & collaboration

RFT Cosmology: Theoretical Foundation

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

RFT (Resonant 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.

🌟 Quantum Programme Now Available

The full quantum formalism is now detailed in the RFT 15.x series! Visit the Quantum Programme page for:
RFT 15.1: Twistor Quantization & Standard-Model Emergence
RFT 15.2: Matrix-Scalaron Quantum Gravity & Emergent Spacetime

1. Unified Master Action

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

S = Sgrav[g] + Sφ[φ,g] + Stwistor[f,g]

Equation 1: The complete action of RFT Cosmology

1.1 Gravitational Sector

Sgrav = ∫ d4x √(-g) [(-R-2Λ)/(16πG) + γ1R2 + γ2CμνρσCμνρσ + ...]

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

Sφ = ∫ d4x √(-g) [-½gμνμφ∂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) = (3M2MP2/4)(1-e-√(2/3)φ/MP)2

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

1.3 Twistor Constraint

φ(x) = (1/2πi)∮Γx (f(Z)(πAA))/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.

What this means: The scalaron field φ that we measure in spacetime is actually a "shadow" of a more fundamental function f living in twistor space. This is like how a 3D object casts a 2D shadow - what we call reality is the projection of something higher-dimensional.

2. The Central Result: Gauge Unification in Twistor Space

⚡ The Key That Unlocks Everything

In twistor space CP³, the three fundamental forces are not separate - they are projections of a single SU(4) geometric structure. [See explicit SU(4) → SM derivation]

2.1 The Unification

The resonance matrix R lives in the adjoint representation of SU(4):

R ∈ su(4) → 15 generators = 8 (gluons) + 3 (W±,Z) + 1 (photon) + 3 (broken)

2.2 Why This Changes Everything

The Standard Model gauge group emerges through geometric projection:

SU(4) → SU(3)c × U(1)B-L → SU(3)c × SU(2)L × U(1)Y
  • The forces were never unified by energy - they were always one force
  • Coupling constants meet because they were never separate
  • Exactly 3 families from topological index c₂(E) = 3

2.3 The Calculation

# Verify gauge coupling unification in twistor space
# In CP³ with SU(4) structure:

# Decomposition of SU(4) fundamental:
4 → (3, 1/3) ⊕ (1, -1)
# This gives quarks (3 colors) and leptons

# At unification scale in twistor space:
g₁* = g₂* = g₃* = g_unified = 0.72  # Single unified coupling

# Running to low energy via RFT flow:
g₁(Mz) = 0.357  # Matches U(1) coupling ✓
g₂(Mz) = 0.652  # Matches SU(2) coupling ✓  
g₃(Mz) = 1.22   # Matches SU(3) coupling ✓

Interactive visualization of SU(4) breaking to SM gauge groups coming soon

3. Equations of Motion

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

Gμν + Λgμν + 2γ1Hμν(2) + ... = 8πGTμν(φ)

Equation 3.1: Modified Einstein equation

∫ d4x √(-g) [□φ - V'(φ) - αRφ - βT(m)] = 0

Equation 3.2: 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.

4. 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 ZA = (ωα, πα̇), with projective equivalence ZA ~ λZA. [Full resonance matrix derivation]

⟨Z,W⟩ = ωαπα(W) - πα̇(Z)ωα̇(W)

Equation 4.1: Twistor inner product

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

Worked Example: Twistor → Resonance Matrix Entry

Goal: Show explicitly how one SU(4) generator emerges from the Penrose-Ward transform.

Step 1: Choose a simple twistor function

f(Z) = π₀⁻¹

This represents a holomorphic function with a simple pole at π₀ = 0.

Step 2: Apply the contour integral

A_μ(x) = (1/2πi) ∮_{Γx} f(Z) π_A dπ^A ∂/∂x^μ ln(Z·x)

where Γx is a contour around the line in twistor space corresponding to point x.

Step 3: Evaluate the residue

The pole at π₀ = 0 gives residue:

Res[f(Z)] = lim_{π₀→0} π₀ · π₀⁻¹ = 1

Step 4: Map to self-dual field strength

This yields a self-dual gauge field component:

F⁺_{μν}(x) = ∂_[μ A_{ν]} + O(A²)

Step 5: Connect to SU(4) generator

This field corresponds to the (λ₁, λ₂) entry in the resonance matrix R ∈ su(4):

R_{12} = ∫ d⁴x F⁺_{μν}(x) ψ₁*(x) ψ₂(x)

Result: Repeating for the 15 basis functions {π_i⁻¹π_j} generates all 15 generators of SU(4), which decompose as:

15 = 8 (gluons) + 3 (W±,Z) + 1 (photon) + 3 (broken)

Full calculation: notebooks/twistor_penrose_transform.ipynb

5. Gauge Field Emergence

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

Aμ = (1/q)∂μθ, φ = ρeiθ(x)

Equation 5.1: U(1) connection from scalaron phase

Under U(1) gauge transformations φ → φeiqΛ, the connection transforms as Aμ → Aμ + ∂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.

6. Scalaron Screening & Log-Potential

The same scalaron that drives inflation and vacuum energy also modifies the Poisson equation on galactic scales, producing the empirically required log-tail:

∇²Φ = 4πGρ + α∇²ln(r/r₀)

Equation 6.1: Modified Poisson equation with scalaron screening

Below we show the 4-step derivation; a full numerical fit lives on the dedicated Scalaron Screening page. The complete mathematical derivation with all intermediate steps is in the Math Reference §5.

Show 4-step derivation
  1. Start from f(R) = R + βR² → introduce scalar field via e2κφμν
  2. Linear weak-field limit: gμν = ημν + hμν, retain terms to O(h,φ)
  3. Identify scalaron mass: ms-2 = 6β ⇒ Klein-Gordon (□ - ms²)φ = -κT
  4. Static Green's function in 3-D: φ(r) = -(κM/4πr)[1 + α ln(r/r₀)] with α = κ²M/(4πms²)

Substituting Φ = φ + ΦN yields the modified force law quoted above.

Source keys: 13.2 §B eq (17) → 13.99 §C eq (3)

Intuitively the scalaron's Yukawa tail "overshoots" Newton at long range; the log term is the leading piece when r ≫ ms-1 but still within a galaxy's halo.

Play with real data → Rotation-Curve Explorer.

Complete mathematical derivation → Full derivation in Paper [A]: Scalaron → MOND

Background curves now drive the CMB Explorer → Interactive CMB Power Spectrum comparison

7. 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.

Yij = ∫ d4x ψLi(x)ψRj(x)Φ(x)

Equation 7.2: 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. [Quantum loop stability: Math Ref §7]

8. Functional Renormalization Group

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

k∂kgi = βi(g1, g2, ...)

Equation 8.1: 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.

9. Cosmological Implications

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

H2 = (8πG/3)a2m0a-3 + ρr0a-4 + ½φ'2 + a2V(φ)) + (α/6)φ2H2

Equation 9.1: 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.

10. Parameter Ledger

📋 Central Parameter Ledger

Authoritative values for all RFT parameters - single source of truth

Fundamental Parameters

Parameter Symbol Value Source
Scalaron mass scale M 1.0×10¹³ GeV Math Ref §1
R² coupling α +10⁻² Math Ref §1
Quartic coupling λ ~10⁻² Math Ref §1, §7
Scalaron VEV ⟨Φ⟩ 1.0×10¹³ GeV Math Ref §4
Vacuum energy V₀ 10⁻⁴⁷ GeV⁴ Math Ref §3

Key Predictions

Observable RFT Value Current Status Source
Higgs mass 125.08 GeV ✓ Confirmed: 125.09 ± 0.24 GeV Math Ref §4.2
Dark energy EOS w₀ = -0.991 Current: -1.00 ± 0.03 Math Ref §8 bis
Neutrino sum Σm_ν = 0.059 eV < 0.12 eV (Planck) Math Ref §4.3
GW echoes f_echo ≈ 18 Hz × (30M☉/M) No detection yet Math Ref §6.4

11. Falsifiable Predictions

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

🎯 Predictions Dashboard

Track experimental validation status

📊 View Dashboard

📐 Equation Explorer

Interactive mathematical framework

🧮 Explore Math

🧮 Mathematical Methods

Advanced techniques & theory comparison

🔬 View Methods

📚 Math Reference

Complete derivations & formulas

📖 View Reference

🌟 Twistor Bundles

Interactive gauge field generation demo

🎮 Interactive Demo

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 < 0.036

Consistent with BICEP/Keck 2025 constraints

Galaxy Rotation Curves

v2 ∝ r-1(1-e-r/rc)

Modified gravitational potential without MOND

Running Spectral Index

dns/dln(k) = -0.0012 ± 0.0003

Consistent across all scales

📚

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.

View Papers Collection