1. Twist Field Tensor TμνTμν and Divergence ψψ
Let the twist field be represented by a rank-2 antisymmetric tensor TμνTμν, encoding yaw, pitch, and roll components.
Define the twist divergence scalar field ψψ as:ψ=∇μTμνuνψ=∇μTμνuν
Where:
- ∇μ∇μ is the covariant derivative (compatible with the background metric)
- uνuν is the comoving observer’s 4-velocity
- ψψ measures net twist flux per unit volume in the observer frame
2. Void Regions as ψ-Minima
Define void boundaries as surfaces where the directional derivative of ψ changes sign:∂iψ=0,∂i2ψ>0∂iψ=0,∂i2ψ>0
These mark local minima of twist divergence — the recoil-dominant phase.
3. Twist Parity Inversion (τₚ)
Introduce a discrete topological marker τp∈{+1,−1}τp∈{+1,−1} representing twist parity (clockwise or counterclockwise coiling):
- In dense regions (e.g., galaxies): τp=+1τp=+1
- In voids: τp=−1τp=−1
Define transition boundaries by:Δτp=2across a ψ-bounded interfaceΔτp=2across a ψ-bounded interface
These correspond to twist confinement breakdown and recoil dominance — a signature of large-scale void formation.
II. Predictive Consequences for Voids
A. Void Shape and Alignment
- Prediction 1: Voids should exhibit anisotropic alignment in the ψ₃-mode field at angular separations corresponding to multipoles ℓ≈3–5ℓ≈3–5, measurable via CMB residual correlations.⟨ψiψj⟩∼cos(Δθij)for disjoint voids at large separation⟨ψiψj⟩∼cos(Δθij)for disjoint voids at large separation
- Test: Cross-correlation between VEGA void centers and low-ℓ Planck ψ₃-residual maps
B. Minimum Energy Confinement and Void Radii
- Define a twist energy functional:
Etwist=∫V[12(∇⋅T)2+V(τp)]d3xEtwist=∫V[21(∇⋅T)2+V(τp)]d3x
Where V(τp)V(τp) is a symmetry-breaking potential favoring τₚ = ±1 in complementary regions.
- Prediction 2: Voids will stabilize at radii corresponding to the first node in ∇⋅T∇⋅T eigenmodes:
Rvoid∼nπkψwith n=1Rvoid∼kψnπwith n=1
- Test: Void size distributions should cluster around stable node scales in the ψ Laplacian spectrum
C. Void–Void Resonant Coupling
- Prediction 3: Voids exhibit collective resonance modes under twist-wave propagation (ψ field oscillations), leading to subtle gravitational lensing echoes:
δψ(t)∝∑nAncos(ωnt+ϕn)δψ(t)∝n∑Ancos(ωnt+ϕn)
- Test: Look for persistent harmonic modulations in weak lensing spectra at frequencies tied to the ψ-twist eigenfrequencies
About the Author
