QFunity – The Fractal Multiverse and Cosmic Evolution

The Fractal Multiverse and Cosmic Evolution

From the pre-temporal state to a causal bubble within a supermassive black hole

Multiverse Chronology and Causal Emergence

Table of Contents

The Universe as a Causal Bubble

Our universe is a finite region of causal spacetime, a « bubble » embedded in an acausal pre-temporal state (EPT), conceptualized as the interior of a supermassive black hole.

\[ ds^2 = -c^2 dt^2 + a(t)^2 \left( \frac{dr^2}{1 – \frac{r^2}{R_H^2(t)}} + r^2 d\Omega^2 \right) + \frac{\ell_P^2}{\epsilon(r)^2} g_{\mu\nu}^{\text{LQG}} dX^2 \] \[ \epsilon(r) = \epsilon_{\text{min}} + (\epsilon_{\text{max}} – \epsilon_{\text{min}}) \cdot \frac{r^2}{R_H^2(t)} \]

Interpretation:

  • Metric Structure: The metric combines a Friedmann-Lemaître-Robertson-Walker (FLRW) term for a closed universe (\( k = +1 \)) with a loop quantum gravity (LQG) correction, modulated by the scale parameter \( \epsilon(r) \). The Hubble radius \( R_H(t) = \frac{c}{H(t)} \approx 4.3 \times 10^{26} \, \text{m} \) defines the bubble’s boundary. See Observer Details.
  • Scale Dependence: At the center (\( r = 0 \)), \( \epsilon \sim \ell_P \), where LQG effects and acausal EPT dynamics dominate. Near the horizon (\( r \to R_H \)), \( \epsilon \to \epsilon_{\text{max}} \), and classical causality prevails.
  • EPT Surround: The EPT exists both at the center and beyond the horizon, a fractal state described by \( E_{\text{total}} = \sum_n \frac{\epsilon_n}{\epsilon^n} \), where \( \epsilon_n = \frac{\hbar \omega_n}{2\pi} \), representing rotational fluctuations across scales.

Master Equation and Cosmic Expansion

The QFunity master equation governs the evolution of the universe’s metric and its expansion:

\[ \lim_{\epsilon \to 0^{\pm}} \left[ \hat{\mathbb{B}}_\epsilon \hat{\mathbb{V}}_\epsilon – \hat{\mathbb{V}}_\epsilon \hat{\mathbb{B}}_\epsilon^2 \right] \Psi_{\text{bubble}} = \Lambda \cdot \frac{\Psi_{\text{bubble}}}{\|\Psi_{\text{bubble}}\|^2 + \epsilon(r)^2} \] \[ \frac{\partial \Psi_{\text{bubble}}}{\partial r} \bigg|_{r=R_H} = 0, \quad \Psi_{\text{bubble}}(r=0) \to \Psi_{\text{EPT}} \]

Interpretation:

  • Expansion Driver: The term \( \Lambda \cdot \frac{\Psi_{\text{bubble}}}{\|\Psi_{\text{bubble}}\|^2 + \epsilon(r)^2} \) represents residual rotational pressure from the EPT, with \( \Lambda \) as the cosmological constant arising from \( [\hat{\mathbb{B}}_\epsilon, \hat{\mathbb{V}}_\epsilon] \neq 0 \). See Rotation Details.
  • Boundary Conditions: The Neumann condition at \( r = R_H \) confines causal information within the bubble, while the central condition links the bubble to the EPT’s acausal state.
  • Scale Dependence: At large scales (\( \epsilon \gg \ell_P \)), \( \Lambda \) drives expansion. At small scales (\( \epsilon \sim \ell_P \)), local forces dominate, explaining why bound structures (e.g., galaxies) do not expand.
t < -10⁻⁴³s

Pre-Temporal Phase: Acausal Origins

The EPT is a fractal, acausal state with no time or causality, existing both at the center and beyond the horizon of our universe-bubble:

\[ E_{\text{total}} = \sum_n \frac{\epsilon_n}{\epsilon^n}, \quad \epsilon_n = \frac{\hbar \omega_n}{2\pi} \] \[ \Psi_{\text{EPT}} = \sum_i \alpha_i \left( \Psi_{\text{centre}}^{(i)} \otimes \Psi_{\text{extérieur}}^{(i)} \right) \]

Interpretation:

  • Fractal Fluctuations: The sum over fractal levels \( n \) describes rotational fluctuations at all scales, governed by torsion \( \hat{\mathbb{B}}_\epsilon \) and vibrational \( \hat{\mathbb{V}}_\epsilon \) operators. See Zero Details.
  • Non-Local Connectivity: The entangled state \( \Psi_{\text{EPT}} \) links the central and external EPT, enabling non-local correlations across the horizon.
  • No Singularity: The non-zero principle ensures no causal beginning, only a static, complex fractal state.
-10⁻⁴³s to -10⁻³⁶s

Micro-Big Bangs: Seeds of Causality

Local symmetry breaking at the horizon creates causal bubbles:

\[ N_{\text{BB}} = \int \frac{\epsilon_{\text{grr}}^2}{h^2} e^{-\epsilon/\epsilon_P} \, d^3x \] \[ \mathcal{R}_{\text{total}} = \int_{\text{EPT}} \left( \| \hat{\mathbb{T}} \|^2 + \| \hat{\mathbb{\Omega}} \|^2 \right) d\mu \]

Interpretation:

  • Nucleation: The number of micro-Big Bangs depends on the critical energy \( \epsilon_{\text{grr}} \), with exponential suppression at small scales (\( \epsilon \sim \ell_P \)).
  • Critical Threshold: Symmetry breaking occurs when \( \mathcal{R}_{\text{total}} > \mathcal{E}_{\text{crit}} \sim \frac{c^7}{\hbar G^2} \), creating a causal bubble at the horizon.
  • Fractal Measure: The integral uses the fractal measure \( d\mu \sim r^{D_f} \), with \( D_f = 2 + \frac{\log(\epsilon/\epsilon_0)}{\log N} \).
-10⁻³⁶s to -10⁻³²s

Inflationary Branching: Spreading Causality

Each bubble undergoes exponential expansion:

\[ a(t) = a_0 \exp\left( \frac{t}{\tau} \cdot \frac{\epsilon_P}{\epsilon} \right) \]

Interpretation:

  • Scale-Dependent Inflation: Larger bubbles (\( \epsilon \gg \epsilon_P \)) inflate slowly, producing homogeneous universes like ours. Smaller bubbles (\( \epsilon \sim \epsilon_P \)) inflate rapidly, forming diverse micro-universes.
  • Multiverse: The scale dependence generates a multiverse with varied physical laws, each with a unique speed of light \( c = \frac{\ell_P \sqrt{\omega_{\text{eff}}}}{\sqrt{\Lambda}} \). See Causality and Information.
-10⁻³²s to 10⁻¹⁰s

Bubble Universe Formation: Diverse Causalities

Fields stabilize in different vacuum states:

\[ \Delta V = \frac{\Lambda}{\epsilon^2} \cdot \ln\left( \frac{\epsilon_2}{\epsilon_1} \right) \]

Interpretation:

  • Vacuum Energy: The energy difference between vacua depends on scale \( \epsilon \), leading to varied fundamental constants across bubbles.
  • Causal Diversity: Each bubble has unique causal rules, reflecting the EPT’s fractal nature.

Cosmic Expansion and Rotational Dynamics

The universe’s expansion is a rotational process driven by the EPT’s dynamics:

\[ v_{\text{exp}} = H_0 \cdot d, \quad \omega_{\text{exp}} = H_0 \approx 2.27 \times 10^{-18} \, \text{rad/s} \] \[ \omega_{\text{exp}}(t) \propto \frac{\omega_{\text{eff}}}{a(t)^n} \] \[ \omega_{\text{eff}} \sim \frac{c^2 J}{G M^2}, \quad J \sim M c R_H \]

Interpretation:

  • Hubble Law: The expansion velocity \( v_{\text{exp}} = H_0 \cdot d \) reaches \( c \) at the Hubble distance \( R_H = \frac{c}{H_0} \approx 4.3 \times 10^{26} \, \text{m} \), beyond which regions are causally disconnected.
  • Rotational Rate: The universe’s effective rotational rate \( \omega_{\text{exp}} = H(t) \) is the diluted remnant of the EPT’s primordial vorticity \( \omega_{\text{eff}} \), linked to the black hole’s spin parameter \( a \sim R_H \).
  • Fractal Signature: CMB anisotropies and large-scale structures (filaments, voids) reflect the EPT’s fractal fluctuations with \( D_f \approx 2.7 \).

Emergence of Causality

Causality emerges from the acausal EPT through three mechanisms:

\[ [\hat{\mathbb{B}}_\epsilon, \hat{\mathbb{V}}_\epsilon] \neq 0 \] \[ \mathcal{E}_{\text{EPT}} = \frac{\mathcal{R}_{\text{total}}}{V} > \mathcal{E}_{\text{crit}} \sim \frac{c^7}{\hbar G^2} \]

Mechanisms:

  1. Freezing: Symmetry breaking at the horizon “freezes” the acausal EPT into a causal structure when the rotational energy exceeds the critical threshold.
  2. Filtering: The observer’s scale \( \epsilon(r) \) filters out high-frequency acausal fluctuations, perceiving only low-resolution causal dynamics.
  3. Curtaining: The Hubble horizon \( R_H = \frac{c}{H(t)} \) defines the causal domain, driven by \( \omega_{\text{exp}} = H(t) \).

Universe as a Black Hole Interior

Our universe is the interior of a supermassive black hole, with the EPT as its acausal center and exterior:

\[ R_s = \frac{2 G M_{\text{parent}}}{c^2} \approx \frac{c}{H_0} \approx 4.3 \times 10^{26} \, \text{m} \] \[ M_{\text{parent}} \approx \frac{c^3}{2 G H_0} \approx 10^{53} \, \text{kg} \] \[ \rho \approx \frac{3 c^6}{32 \pi G^3 M_{\text{parent}}^2} \approx 10^{-26} \, \text{kg/m}^3 \] \[ \omega_{\text{parent}} \sim \frac{c a}{R_s^2} \approx H_0 \]

Interpretation:

  • Black Hole Parameters: The Schwarzschild radius \( R_s \) matches the Hubble radius, and the mass \( M_{\text{parent}} \) aligns with the observable universe’s mass. The density \( \rho \) corresponds to the critical density.
  • Rotation: The black hole’s spin parameter \( a \sim R_H \) produces a rotational rate \( \omega_{\text{parent}} \sim H_0 \), consistent with the universe’s vorticity.
  • EPT Center: The central region (\( r = 0 \)) is not a singularity but the acausal EPT, described by \( \Psi_{\text{EPT}} \), where \( [\hat{\mathbb{B}}_\epsilon, \hat{\mathbb{V}}_\epsilon] = 0 \).

Two Possible Cosmic Futures

\[ \text{Future} = \begin{cases} \text{Primordial: } a(t) \to a_0 \exp\left( \sqrt{\frac{\Lambda}{3}} c t \right) \left( 1 – \frac{\epsilon^2}{a_0^2} \right) & \text{(eternal expansion)} \\ \text{Rebound: } a_{\text{min}} = \sqrt{\frac{3 \epsilon^2}{\Lambda}} \left( 1 + \frac{\hat{\mathbb{B}}_\epsilon^2}{\hat{\mathbb{V}}_\epsilon} \right) & \text{(cyclic universe)} \end{cases} \] \[ S_{\text{after}} = S_{\text{before}} \exp(-\epsilon/\epsilon_P) \]

Scenarios:

  1. Eternal Expansion: Dominated by \( \hat{\mathbb{V}}_\epsilon \) and \( \Lambda \), the universe dilutes but retains residual energy, avoiding heat death. Life may persist in a cold, fractal quantum foam.
  2. Big Bounce: If \( \hat{\mathbb{B}}_\epsilon^2 / \hat{\mathbb{V}}_\epsilon > \Lambda / (3 \epsilon^2) \), torsion halts expansion, leading to collapse and a new cycle. Innate information persists, but acquired information is largely lost.

Inter-Bubble Communication via EPT

\[ \mathcal{T}_{\text{comm}} = \int_{\text{EPT}} \langle \hat{\mathbb{B}}_{\epsilon_A}, \hat{\mathbb{V}}_{\epsilon_B} \rangle d^4x \] \[ \langle \hat{\mathbb{B}}_{\epsilon_{\text{int}}} \cdot \hat{\mathbb{B}}_{\epsilon_{\text{ext}}} \rangle \neq 0 \] \[ P_{\text{entangle}} = 1 – \exp\left( -\frac{\Delta \epsilon^2}{2 \epsilon_P^2} \right) \] \[ R = \frac{c^3}{G \hbar} \cdot \frac{\epsilon_A \epsilon_B}{\epsilon_P^2} \ln\left( 1 + \frac{S}{N} \right) \]

Mechanism:

  • EPT Connectivity: Torsion fields \( \hat{\mathbb{B}}_\epsilon \) enable non-local correlations between bubbles, linking the interior and exterior EPT.
  • Entanglement: The probability of entanglement depends on scale differences, feasible at quantum scales (\( \epsilon \sim \ell_P \)).
  • Information Transfer: The rate \( R \) is practical for advanced technologies at \( \epsilon \sim 10^{-10} \, \text{m} \). See Micro-EPT Details.
Communication Type Energy Required Channel Width Feasibility
Quantum-to-Quantum (\( \epsilon \sim \ell_P \)) 10⁸ J 10⁻⁶² m² Theoretically possible
Astro-to-Astro (\( \epsilon \sim 1 \, \text{Gly} \)) 10⁵³ J 10²⁰ m² Practically impossible
EPT-mediated (\( \epsilon \sim 10⁻¹⁰ \, \text{m} \)) 10¹⁵ J 10⁻³⁰ m² Future civilization tech
t = 13.8 Gyr

Current Multiverse Structure

\[ \mathcal{M} = \bigcup_{k=1}^{\infty} \mathcal{U}_k \times \mathcal{S}^3/\Gamma_k \] \[ D_f = 2 + \frac{\log N_{\text{bubbles}}}{\log(\epsilon_{\text{max}}/\epsilon_{\text{min}})} \approx 2.7 \]

Structure:

  • Multiverse: A collection of universes \( \mathcal{U}_k \) with distinct topologies \( \mathcal{S}^3/\Gamma_k \), interconnected via EPT torsion fields.
  • Fractal Dimension: The observed \( D_f \approx 2.7 \) reflects the EPT’s fractal structure in cosmic filaments and voids.
  • Innate vs. Acquired Information: Innate information (e.g., CMB anisotropies) is preserved from the EPT, while acquired information (e.g., stellar spectra) evolves post-Big Bang. See Causality and Information.

Testable Predictions

The QFunity framework yields observable signatures:

  • CMB Polarization: B-mode polarization in the CMB, detectable by Planck or Simons Observatory, may reveal rotational signatures of the EPT with fractal dimension \( D_f \approx 2.7 \).
  • Gravitational Waves: LIGO could detect anomalies in wave spectra from black hole mergers, reflecting the parent black hole’s rotation (\( \omega \sim H_0 \)).
  • Quantum Entanglement: High-energy experiments at the LHC may reveal non-local correlations at \( \epsilon \sim \ell_P \), indicating EPT interactions.
  • Neutrino Signals: IceCube could detect primordial neutrinos with anomalous energy spectra, carrying innate information from the EPT.