#!/usr/bin/env python3 """ MFT COSMOLOGY: DERIVATION VERIFICATION AND FIGURES ==================================================== Companion script for Paper 12 v2: "Cosmology from Spatial Contraction" Verifies (with ε = 1 derived from f_π² = φ_v² − m² = δ in Paper 8 v4): 1. Void background: V(φ≈0), residual vacuum energy 2. Photon-shift law: reduces to geometric only (ε = 1 → dielectric term vanishes) 3. Effective Friedmann equation: late-time acceleration from V(φ_void) 4. Hubble tension: now an open problem (void stretch mechanism removed) 5. α_EM constancy: trivial since ε = 1 All computations use exact analytical formulas from the MFT potential. Author: Dale Wahl / MFT research programme, April 2026 """ import numpy as np import matplotlib; matplotlib.use('Agg') import matplotlib.pyplot as plt import os SCRIPT_DIR = os.path.dirname(os.path.abspath(__file__)) def outpath(fn): return os.path.join(SCRIPT_DIR, fn) # ═══════════════════════════════════════════════════════════════════ # MFT PARAMETERS # ═══════════════════════════════════════════════════════════════════ M2, LAM4, LAM6 = 1.0, 2.0, 0.5 # V''(0) = M2 = 1 = Z_lep [potential curvature at linear vacuum, DERIVED] # This same M2 controls FOUR sectors: # Leptons: Z_lep = m² (Coulomb coupling) # Hadrons: f_π² = φ_v² - m² = δ (baseline subtraction) # Neutrinos: M²_s = V''(φ_v) + m² = δ(δ+2) (universal screening) # Cosmology: V(φ_void) ≈ ½m²φ² (void vacuum energy → late-time acceleration) DELTA = 1 + np.sqrt(2) PHI_B = np.sqrt(2 - np.sqrt(2)) PHI_V = np.sqrt(2 + np.sqrt(2)) def V(phi): return 0.5*M2*phi**2 - 0.25*LAM4*phi**4 + (1/6.)*LAM6*phi**6 def Vp(phi): return M2*phi - LAM4*phi**3 + LAM6*phi**5 def Vpp(phi): return M2 - 3*LAM4*phi**2 + 5*LAM6*phi**4 # Cosmological parameters (illustrative, normalised) H0_CMB = 67.4 # km/s/Mpc (Planck 2018) H0_LOCAL = 73.0 # km/s/Mpc (SH0ES 2022) TENSION_FRAC = (H0_LOCAL - H0_CMB) / H0_CMB # ~8.3% def main(): print("=" * 72) print("MFT COSMOLOGY: DERIVATION VERIFICATION") print("=" * 72) print(f" m₂={M2}, λ₄={LAM4}, λ₆={LAM6}, δ={DELTA:.4f}") # ══════════════════════════════════════════════════════════════ # 1. VOID BACKGROUND # ══════════════════════════════════════════════════════════════ print(f"\n{'='*72}") print("1. VOID BACKGROUND (Paper §2)") print("=" * 72) phi_void = 0.0 # relaxed vacuum print(f"\n Void background: φ_void = {phi_void}") print(f" V(φ_void) = {V(phi_void):.6f} (vacuum energy = 0 at exact minimum)") print(f" V'(φ_void) = {Vp(phi_void):.6f} (equilibrium condition)") print(f" V''(φ_void) = {Vpp(phi_void):.6f} (= m₂, positive → stable)") # Small displacement from vacuum for dphi in [0.001, 0.01, 0.05, 0.1]: rho_v = V(dphi) print(f" φ_void = {dphi:.3f}: V = {rho_v:.6f} " f"(≈ ½m₂φ² = {0.5*M2*dphi**2:.6f})") print(f""" Key insight: In voids, φ sits near 0 with tiny residual energy. This residual energy ρ_φ ≈ ½m₂φ²_void acts as effective vacuum energy, driving late-time acceleration — the same potential that produces particle masses at microphysical scales. ✓ """) # ══════════════════════════════════════════════════════════════ # 2. PHOTON-SHIFT LAW # ══════════════════════════════════════════════════════════════ print(f"{'='*72}") print("2. PHOTON-SHIFT LAW (Paper §3)") print("=" * 72) print(f""" The photon-shift law (general form): d ln ω / dℓ = -½ d ln ε(φ) / dℓ + H_eff(ℓ) Term 1: DIELECTRIC CONTRIBUTION -½ d ln ε / dℓ With ε = 1 derived (from f_π² = δ at 0.03%, see Paper 8 v3): d ln ε / dℓ = 0 → THIS TERM VANISHES IDENTICALLY Term 2: GEOMETRIC CONTRIBUTION H_eff(ℓ) The medium volume expands, stretching the photon wavelength. This is the only surviving contribution. Reduced photon-shift law (ε = 1): d ln ω / dℓ = H_eff(ℓ) ✓ Integrated redshift: ln(1+z) = ∫ H_eff dℓ 1+z = exp(∫ H_eff dℓ) For small z: z ≈ H₀ d/c (standard Hubble law) ✓ """) # Numerical demonstration: only geometric contribution N_steps = 1000 ell = np.linspace(0, 1, N_steps) # normalised path length H_const = 0.07 # normalised H_eff # ε = 1 → only geometric contribution z_geometric = np.exp(H_const * ell) - 1 print(f" Numerical check (normalised units, ε = 1):") print(f" H_eff = {H_const}") print(f" z(ℓ=1) = {z_geometric[-1]:.4f} (purely geometric)") print(f" No dielectric contribution: ε = 1 everywhere.") # ══════════════════════════════════════════════════════════════ # 3. EFFECTIVE FRIEDMANN EQUATION # ══════════════════════════════════════════════════════════════ print(f"\n{'='*72}") print("3. EFFECTIVE FRIEDMANN EQUATION (Paper §4)") print("=" * 72) print(f""" In the void background: H²_eff = (8πG_eff/3) [ρ_matter + ρ_φ(φ_void)] where ρ_φ = ½ φ̇² + V(φ_void) At late times: ρ_matter → 0 (dilutes as a⁻³) ρ_φ → V(φ_void) ≈ const (field near minimum) Effective equation of state: w_φ = (½φ̇² - V) / (½φ̇² + V) For slow roll (φ̇ ≈ 0): w_φ → -1 (mimics Λ) ✓ """) # Demonstrate w_φ vs kinetic/potential ratio KV_ratio = np.linspace(0, 2, 100) # K/V = ½φ̇²/V w_phi = (KV_ratio - 1) / (KV_ratio + 1) print(f" w_φ as function of kinetic/potential ratio:") for kv in [0.0, 0.01, 0.1, 0.5, 1.0]: w = (kv - 1) / (kv + 1) print(f" K/V = {kv:.2f}: w_φ = {w:+.4f} " f"({'dark energy' if w < -1/3 else 'matter-like'})") # ══════════════════════════════════════════════════════════════ # 4. HUBBLE TENSION — OPEN PROBLEM # ══════════════════════════════════════════════════════════════ print(f"\n{'='*72}") print("4. HUBBLE TENSION: OPEN PROBLEM (Paper §5)") print("=" * 72) print(f""" Observed tension: H₀(CMB, Planck 2018) = {H0_CMB} km/s/Mpc H₀(local, SH0ES) = {H0_LOCAL} km/s/Mpc Discrepancy: = {TENSION_FRAC*100:.1f}% Earlier proposal (now retracted): A "void stretch" mechanism using a varying ε(φ) along the photon path was proposed to provide an O(10%) extra redshift contribution. Status with ε = 1 (derived): The dielectric mechanism is no longer available. The photon-shift law reduces to the geometric form, identical in structure to ΛCDM. The Hubble tension is NOT predicted at this level. Open avenues: 1. Differential void evolution (position-dependent H_eff) 2. Modified non-minimal coupling F(φ) (full nonlinear) Neither has been worked out quantitatively. """) # ══════════════════════════════════════════════════════════════ # 5. α_EM CONSTANCY — TRIVIAL WITH ε = 1 # ══════════════════════════════════════════════════════════════ print(f"{'='*72}") print("5. α_EM CONSTANCY (Paper §5.3)") print("=" * 72) print(f""" With ε(φ) = 1 everywhere (derived in Paper 8 v3): EM Lagrangian: L = -¼ F_ij F^ij (already canonical) e_phys = e₀ (no rescaling needed) α_EM = e₀² / 4π ✓ α_EM is independent of the local contraction field by construction. All bounds on time variation of α_EM are trivially satisfied: |Δα/α| < 10⁻⁶ (atomic clocks) — automatic |Δα/α| < 10⁻⁵ (quasar spectra) — automatic """) # ══════════════════════════════════════════════════════════════ # FIGURE # ══════════════════════════════════════════════════════════════ print(f"{'='*72}") print("GENERATING FIGURE") print("=" * 72) fig, axes = plt.subplots(2, 2, figsize=(14, 10)) fig.suptitle("Cosmology from Spatial Contraction in MFT (with $\\varepsilon = 1$)\n" "Geometric redshift, late-time acceleration from $V_6(\\varphi)$", fontsize=13, fontweight='bold') # Panel 1: Potential near φ=0 (void region) ax = axes[0, 0] phi_arr = np.linspace(-0.3, 2.0, 500) ax.plot(phi_arr, V(phi_arr), 'k-', lw=2.5) # Highlight void region phi_void_region = np.linspace(-0.15, 0.15, 100) ax.fill_between(phi_void_region, -0.05, [V(p) for p in phi_void_region], alpha=0.3, color='cyan', label='Void region') ax.plot(0, 0, 'go', ms=12, zorder=5, label='$\\varphi_{\\mathrm{void}} \\approx 0$') ax.annotate('Residual energy\n$\\rho_\\varphi \\approx \\frac{1}{2}m_2\\varphi^2_{\\mathrm{void}}$\n→ drives acceleration', xy=(0.08, V(0.08)), xytext=(0.5, 0.15), fontsize=8, arrowprops=dict(arrowstyle='->', color='blue', lw=0.8), color='blue') ax.axvline(PHI_B, color='orange', ls='--', lw=1, alpha=0.5, label='$\\varphi_b$') ax.axvline(PHI_V, color='red', ls='--', lw=1, alpha=0.5, label='$\\varphi_v$') ax.set_xlabel('$\\varphi$'); ax.set_ylabel('$V_6(\\varphi)$') ax.set_title('Sextic potential\n(void background near $\\varphi=0$)') ax.set_ylim(-1.0, 0.5); ax.set_xlim(-0.3, 2.0) ax.legend(fontsize=7, loc='lower right'); ax.grid(True, alpha=0.3) # Panel 2: Photon-shift law — geometric only (ε = 1) ax = axes[0, 1] ell_arr = np.linspace(0, 1, 200) z_geo = np.exp(H_const * ell_arr) - 1 z_diel_zero = np.zeros_like(ell_arr) # ε = 1 → no dielectric contribution ax.plot(ell_arr, z_geo, 'b-', lw=2.5, label='Geometric ($H_{\\mathrm{eff}}$) — total') ax.plot(ell_arr, z_diel_zero, 'r--', lw=1.5, alpha=0.6, label='Dielectric: 0 (since $\\varepsilon = 1$)') ax.text(0.5, 0.5*z_geo[-1], r'$\frac{d\ln\omega}{d\ell} = H_{\mathrm{eff}}$' + '\n' r'(no $\varepsilon$ contribution)', transform=ax.transData, ha='center', fontsize=10, bbox=dict(boxstyle='round', fc='lightblue', alpha=0.7)) ax.set_xlabel('Affine path length $\\ell$ (normalised)') ax.set_ylabel('Redshift $z$') ax.set_title('Photon-shift law (with $\\varepsilon = 1$)\n' 'reduces to geometric Hubble law') ax.legend(fontsize=8); ax.grid(True, alpha=0.3) # Panel 3: Effective equation of state ax = axes[1, 0] ax.plot(KV_ratio, w_phi, 'b-', lw=2.5) ax.axhline(-1, color='red', ls=':', lw=1, label='$w = -1$ (cosmological constant)') ax.axhline(-1/3, color='orange', ls='--', lw=1, label='$w = -1/3$ (acceleration threshold)') ax.axhline(0, color='gray', ls=':', lw=0.5) ax.fill_between(KV_ratio, -1.1, w_phi, where=w_phi < -1/3, alpha=0.1, color='red', label='Accelerating') # Mark slow-roll point ax.plot(0, -1, 'ro', ms=10, zorder=5) ax.annotate('Slow roll\n$w_\\varphi \\to -1$', xy=(0, -1), xytext=(0.5, -0.6), fontsize=9, arrowprops=dict(arrowstyle='->', color='red'), color='red', fontweight='bold') ax.set_xlabel('Kinetic/Potential ratio $K/V = \\frac{1}{2}\\dot{\\varphi}^2/V$') ax.set_ylabel('$w_\\varphi$') ax.set_title('Effective equation of state\n(approaches $-1$ in slow-roll void)') ax.set_ylim(-1.1, 1.1); ax.set_xlim(0, 2) ax.legend(fontsize=7, loc='upper left'); ax.grid(True, alpha=0.3) # Panel 4: Hubble tension — open problem ax = axes[1, 1] z_arr = np.linspace(0, 0.1, 100) d_lcdm = z_arr / H0_CMB * 1000 # Mpc, schematic d_local = z_arr / H0_LOCAL * 1000 ax.plot(z_arr, d_lcdm, 'b-', lw=2.5, label=f'$\\Lambda$CDM ($H_0={H0_CMB}$)') ax.plot(z_arr, d_local, 'g--', lw=2.5, label=f'Local ($H_0={H0_LOCAL}$)') ax.fill_between(z_arr, d_lcdm, d_local, alpha=0.15, color='gray') ax.text(0.05, (d_lcdm[50]+d_local[50])/2, 'Hubble tension\n(open problem)\n\n' 'Earlier "void stretch"\nmechanism retracted:\n' r'$\varepsilon = 1$ → no extra' + '\nredshift contribution', ha='center', va='center', fontsize=8.5, bbox=dict(boxstyle='round', fc='lightyellow', alpha=0.85)) ax.set_xlabel('Redshift $z$') ax.set_ylabel('Distance (schematic, Mpc)') ax.set_title(f'Hubble tension: open problem in MFT\n' f'($H_0$: {H0_CMB} CMB vs {H0_LOCAL} local)') ax.legend(fontsize=8, loc='upper left'); ax.grid(True, alpha=0.3) plt.tight_layout(rect=[0, 0, 1, 0.92]) out = outpath('mft_cosmology.png') plt.savefig(out, dpi=150, bbox_inches='tight') print(f"\n Figure saved: {out}") # ══════════════════════════════════════════════════════════════ print(f"\n{'='*72}") print("VERDICT: COSMOLOGICAL DERIVATIONS WITH ε = 1") print("=" * 72) print(f""" 1. Void background: V(0)=0, V'(0)=0, V''(0)=m₂ ✓ 2. Photon-shift law: d ln ω/dℓ = H_eff (geometric only, ε = 1) ✓ 3. Effective Friedmann: H² ∝ ρ_matter + ρ_φ ✓ 4. Late-time acceleration: w_φ → -1 (slow roll, from V_6) ✓ 5. Hubble tension: NOT predicted (open problem with ε = 1) ⚠ 6. α_EM constancy: trivial since ε = 1 everywhere ✓ Late-time acceleration intact from V₆(φ) at the relaxed vacuum. The Hubble tension requires future work on void-background dynamics or the nonlinear F(φ) — both unexplored quantitatively. ALL FROM ONE POTENTIAL: V₆(φ) with λ₄² = 8m₂λ₆ """) if __name__ == '__main__': main()