Cosmological Evolution of the Information Field: Running Complexity Coupling and Unified Cosmological Phases

Author: Nova Spivack
Date: June 2025

Abbreviation: IP.Cosmo

Abstract

This paper develops a unified cosmological model based on the complexity density field ω(x,t) (dimensions [L⁻³]) from Information Physics, incorporating a temperature-dependent running coupling that resolves energy scale hierarchies. The fundamental energy-complexity relationship $dE = \alpha_{\text{eff}}(T)d\Omega$ , with $\alpha_{\text{eff}}(T) = \pi k_B T \times F(T/T_{\text{Planck}})$ and scaling function $F(x) = 1 + A x^n$ ( $A \sim 10^{60}, n \sim 3$ ), provides significant enhancement at Planck temperatures. The ω-field evolves through three natural cosmological phases governed by $V_{\text{total}}(\omega,T) = \alpha_{\text{eff}}(T)\omega + V_{\text{self}}(\omega)$ , where the intrinsic self-interaction $V_{\text{self}}(\omega) = \frac{1}{2}\beta^2\omega^2 - \mu^3\omega + \frac{\lambda_4}{4!}\omega^4$ (with β², μ³, λ₄ having dimensions [EL³], [E], [EL⁹] respectively) creates metastable states. Phase I features inflation driven by the enhanced $\alpha_{\text{eff}}(T_{\text{Planck}})\omega$ term setting the energy scale, in conjunction with a self-interaction term like $\lambda_4\omega^4$ ensuring slow-roll dynamics. Phase II exhibits metastable complexity equilibrium during radiation/matter domination. Phase III presents relaxation-driven dark energy as ω transitions to its true ground state (ω=0) when thermal energy k_BT_cosmic drops below the energy scale characterizing the metastable barrier, defining a critical temperature T_c. This framework naturally explains cosmic acceleration timing and provides falsifiable predictions linking laboratory measurements of IP parameters (κ, β², μ³, λ₄) to cosmological observables via the “IP Consistency Test.” The model unifies inflation and dark energy under a single information-geometric foundation.

Keywords: Information Physics, Complexity Density, Running Coupling, Inflation, Dark Energy, Phase Transitions, Metastable States, Ω-Field Cosmology, Information Geometry.

1. Introduction

1.1 The Energy Scale Hierarchy Challenge in Information Cosmology

Information Physics (IP) establishes the fundamental relationship $dE = \alpha_0 d\Omega$ where $\alpha_0 = \pi k_B T$ links energy changes to information geometric complexity evolution (IP.Found). Applying this directly to cosmology faces a severe energy scale problem: local complexity effects involve energies $\sim k_B T \sim 10^{-21} \text{ J}$ , while cosmological phenomena require energy densities from $\sim 10^{-9} \text{ J/m}^3$ (dark energy) to vastly higher scales for inflation. This paper resolves this by proposing a running complexity-energy coupling.

1.2 Running Complexity Coupling: Information Degrees of Freedom Scaling

We generalize the fundamental relationship to $dE = \alpha_{\text{eff}}(T)d\Omega$ , where the effective coupling is:

$\alpha_{\text{eff}}(T) = \pi k_B T \times F(T/T_{\text{Planck}})$ (1)

The dimensionless scaling function $F(x)$ (where $x = T/T_{\text{Planck}}$ ) encodes how accessible information degrees of freedom vary with energy scale. It must satisfy:

Low-energy limit (x → 0): F(x) → 1, recovering $\alpha_{\text{eff}}(T) \approx \pi k_B T$ .
High-energy scaling (x → 1): F(x) provides significant enhancement. A phenomenological form is:

$F(x) = 1 + A \cdot x^n$ (2)

where $A \sim 10^{60} - 10^{66}$ and $n \sim 2-3$ are dimensionless constants. This enhancement arises because as energy density approaches Planckian scales, quantum vacuum complexity, virtual particle degrees of freedom, and spacetime geometry fluctuations contribute vastly more to the accessible information structure.

1.3 Unified Three-Phase Cosmological Evolution

This paper demonstrates that a single complexity density field ω(x,t), governed by this running coupling and an intrinsic self-interaction potential, naturally evolves through three distinct cosmological phases, potentially unifying inflation and dark energy.

2. The ω-Field Theory with Running Coupling

2.1 Field Lagrangian and Fundamental Dynamics

The complexity density field ω(x,t) (dimensionless complexity per unit volume, dimensions [L⁻³]) evolves according to the Lagrangian density:

$L_{\omega} = \frac{1}{2}\kappa (g^{\mu\nu}\partial_{\mu}\omega \partial_{\nu}\omega) - V_{\text{total}}(\omega, T_{\text{cosmic}}(t))$ (3)

where κ is the universal kinetic coefficient (dimensions [EL⁵]) from IP.Field. The total potential combines the direct thermal energy term and intrinsic self-interactions:

$V_{\text{total}}(\omega, T_{\text{cosmic}}) = \alpha_{\text{eff}}(T_{\text{cosmic}}(t)) \omega + V_{\text{self}}(\omega)$ (4)

2.2 Self-Interaction Potential and Metastable Structure

The intrinsic self-interaction potential V_self(ω) is crucial for creating metastable states and driving late-time dynamics. We propose a form:

$V_{\text{self}}(\omega) = V_0 + \frac{1}{2}\beta^2\omega^2 - \mu^3\omega + \frac{\lambda_4}{4!}\omega^4$ (5)

where V₀ is a baseline vacuum energy. The parameters have dimensions: β²: [EL³]; μ³: [E]; λ₄: [EL⁹]. These ensure each term in V_self(ω) is an energy density, given ω is [L⁻³]. With appropriate signs (e.g., β² > 0, μ³ > 0 to create a barrier/dip, λ₄ > 0 for stability at large ω), this potential can exhibit a true minimum (e.g., at ω=0 if V₀ and other terms are set accordingly) and a local, metastable minimum at ω_meta > 0.

2.3 Field Evolution Equation

The equation of motion for homogeneous ω(t) in FLRW spacetime ( $ds^2 = -c^2 dt^2 + a(t)^2 d\mathbf{x}^2$ ) is:

$\kappa(\ddot{\omega} + 3H\dot{\omega}) + c^2 \frac{\partial V_{\text{total}}}{\partial \omega} = c^2 J_{\text{source}}(t)$ (6)

where H = ȧ/a. Assuming T_cosmic is primarily a function of a(t) and $\partial T_{\text{cosmic}}/\partial \omega \approx 0$ for the purpose of the potential derivative, then:

$\frac{\partial V_{\text{total}}}{\partial \omega} = \alpha_{\text{eff}}(T_{\text{cosmic}}) + \beta^2\omega - \mu^3 + \frac{\lambda_4}{3!}\omega^3$ (7)

The source term J_source(t) (dimensions [EL⁻³]) represents net cosmic complexity generation/destruction not captured by V_total (e.g., from ongoing structure formation).

2.4 Stress-Energy Tensor

The ω-field contributes energy density ρ_ω and pressure p_ω:

$\rho_{\omega} = \frac{1}{2}\frac{\kappa}{c^2}\dot{\omega}^2 + V_{\text{total}}(\omega, T_{\text{cosmic}})$ (8)

$p_{\omega} = \frac{1}{2}\frac{\kappa}{c^2}\dot{\omega}^2 - V_{\text{total}}(\omega, T_{\text{cosmic}})$ (9)

3. Phase I: Enhanced-Coupling Inflation (T_cosmic ~ T_Planck)

3.1 Inflationary Energy Scale from Running Coupling

At T_cosmic ≈ T_Planck, the coupling $\alpha_{\text{eff}}(T_{\text{Planck}}) \approx \pi k_B T_{\text{Planck}} \times A$ (with $A \sim 10^{60}$ ) is enormously enhanced. If the early universe possesses an initial complexity density ω_initial (potentially related to ω_Planck or arising from quantum vacuum complexity), the term α_eff(T_Planck)ω_initial in V_total can provide the dominant energy density $\rho_{\text{inf}} \sim M_{\text{Pl}}^4$ required for inflation.

3.2 Inflationary Dynamics

Inflation is driven by the effective potential during this epoch. If ω starts at a large value, terms like $(\lambda_4/4!)\omega^4$ in V_self(ω) can provide a sufficiently flat plateau for slow-roll inflation, with the overall energy scale set by the α_eff(T_cosmic)ω term. As ω rolls down this potential, or as T_cosmic evolves, inflation proceeds.

3.3 Slow-Roll Parameters and Observational Predictions

The slow-roll parameters are (M_Pl² = c⁴/(8πG)):

$\epsilon_V = \frac{M_{Pl}^2 c^4}{2\kappa} \left( \frac{\partial V_{\text{eff}}/\partial\omega}{V_{\text{eff}}} \right)^2$ (10)

$\eta_V = \frac{M_{Pl}^2 c^4}{\kappa} \left( \frac{\partial^2 V_{\text{eff}}/\partial\omega^2}{V_{\text{eff}}} \right)$ (11)

where V_eff is the dominant part of V_total during inflation. The scalar spectral index is $n_s \approx 1 - 6\epsilon_V + 2\eta_V$ . The tensor-to-scalar ratio is $r \approx 16\epsilon_V (\kappa / (M_{Pl}^2 c^4))$ if κ is the coefficient of a canonically normalized kinetic term for ω (this relation for `r` needs careful checking for non-canonical κ). These are testable against CMB data.

3.4 Natural End of Inflation

Inflation terminates as T_cosmic drops (reducing α_eff(T)) and/or ω rolls into a steeper part of V_self(ω), violating slow-roll conditions. The ω-field then reheats the universe through its (direct or indirect) couplings to Standard Model particles.

4. Phase II: Metastable Complexity Equilibrium

4.1 Transition to Self-Interaction Dominance

Post-reheating, as T_cosmic drops significantly, F(T/T_Planck) → 1, so $\alpha_{\text{eff}}(T) \rightarrow \pi k_B T_{\text{cosmic}}$ . This thermal term becomes subdominant to V_self(ω) (Eq. 5). The field ω then seeks a minimum of V_self(ω).

4.2 Metastable State Characteristics

The potential $V_{\text{self}}(\omega) = V_0 + \frac{1}{2}\beta^2\omega^2 - \mu^3\omega + \frac{\lambda_4}{4!}\omega^4$ , with β²>0, μ³>0, λ₄>0, can be engineered to have a true minimum (e.g., at ω=0 by adjusting V₀) and a local, metastable minimum at ω_meta > 0. The location of ω_meta is found from $\partial V_{\text{self}}/\partial\omega = \beta^2\omega - \mu^3 + (\lambda_4/6)\omega^3 = 0$ . For example, if the μ³ term creates a dip and λ₄ω⁴ provides the subsequent barrier and rise, a metastable state can exist. An approximate location if the cubic term dominates at the minimum over the linear is $\omega_{\text{meta}} \sim \sqrt{6\mu^3/\lambda_4}$ (actual solution from cubic equation).

The ω-field can become trapped in this metastable state ω_meta by Hubble friction during the radiation and matter domination eras, contributing a small, nearly constant energy density.

5. Phase III: Relaxation-Driven Dark Energy

5.1 Critical Temperature for Phase Transition

The transition to dark energy occurs when the universe cools sufficiently for ω to escape the metastable minimum. This is triggered when the thermal energy k_BT_cosmic becomes comparable to the energy scale of the barrier height ΔV_barrier = V_self(ω_{barrier_top}) – V_self(ω_meta), or more simply, when the thermal “tilting” of the potential by the $\alpha_{\text{eff}}(T_c)\omega_{\text{meta}}$ term is no longer sufficient to stabilize ω_meta against the drive towards the true minimum.

A critical temperature T_c can be defined such that for T_cosmic < T_c, the barrier can be overcome. T_c will depend on the parameters of V_self(ω) (β², μ³, λ₄). For instance, a simple estimate might be when $\pi k_B T_c \omega_{\text{meta}}$ (the effective linear drive at low T) is no longer balanced by the restoring forces of the metastable well, or when $k_B T_c$ is comparable to the energy scale defined by μ³ (the parameter controlling the barrier). A more precise derivation would involve analyzing the stability of the ω_meta minimum as T_cosmic drops. For example, if the metastable minimum is approximately $\omega_{\text{meta}} \sim \mu^3/\beta^2$ (neglecting λ₄ for the minimum’s location), then T_c might scale as:

$T_c \sim \frac{\mu^3}{\pi k_B}$ (12)

This makes T_c dependent on the fundamental energy scale μ³ from V_self.

5.2 Dark Energy Equation of State

As ω slowly rolls from ω_meta towards the true minimum (e.g., ω=0), its kinetic energy $\dot{\omega}^2$ is small. The energy density $\rho_{\omega} \approx V_{\text{total}}(\omega, T_{\text{cosmic} \approx 0}) \approx V_{\text{self}}(\omega)$ and pressure $p_{\omega} \approx -V_{\text{self}}(\omega)$ . This yields an equation of state:

$w_{\omega}(z) \approx -1 + \frac{\kappa \dot{\omega}^2/c^2}{V_{\text{self}}(\omega)} \approx -1 + \delta w(z)$ (13)

where δw(z) > 0 is small and evolves as ω rolls, reflecting the dynamics of the phase transition.

5.3 Resolution of the Coincidence Problem

The phase transition mechanism naturally explains why dark energy becomes important recently. T_c is determined by fundamental IP parameters (β², μ³, λ₄). When T_cosmic drops below T_c (observed to be around z ~ 0.5-1), the relaxation phase begins, causing ρ_ω to become comparable to matter density.

6. The Enhanced IP Consistency Test

The model’s strength lies in linking parameters across scales:

Local IP Parameters: κ, and the parameters of V_self(ω) (β², μ³, λ₄) can be constrained by laboratory experiments measuring complexity-energy relationships ( $dE = \pi k_B T_{\text{local}} d\omega$ ), complexity wave propagation (IP.Field), and ψ-quanta interactions (IP.Quantum).
Cosmological Observables: The *same* κ and V_self(ω) parameters, along with the running coupling parameters A and n in F(T/T_Planck), must fit inflationary data (n_s, r), the dark energy onset timing (T_c), and w_ω(z).

A consistent set of {κ, β², μ³, λ₄, A, n} explaining both local IP phenomena and cosmic history would strongly validate the theory. Discrepancies, or the need for vastly different parameter sets for local vs. cosmic phenomena beyond what the running coupling F(x) can account for, would falsify or severely constrain it.

7. Observational Signatures and Predictions

7.1 Cosmic Microwave Background (CMB)

Inflationary Signatures: A distinct n_s-r relationship predicted by the α_eff(T)ω + V_self(ω) inflationary potential. The running of α_eff(T) during inflation introduces a temperature-dependent evolution that could yield unique features in the primordial power spectrum.
Late-time ISW Effect: The phase transition driving dark energy will cause V_total to evolve, leading to a characteristic ISW signal. The shape and amplitude of this signal depend on the specifics of the ω-field rolling from ω_meta.
CMB Lensing: The evolving ω-field contributes to the lensing potential, potentially altering the CMB lensing spectrum in ways distinguishable from ΛCDM.

7.2 Large-Scale Structure (LSS)

Modified Growth History: The growth rate of structures, f(z)σ₈, will be affected by the dynamic w_ω(z) during the dark energy phase transition.
Baryon Acoustic Oscillations (BAO): The BAO scale provides a standard ruler; its apparent evolution can test the expansion history predicted by the ω-field model.
Void Statistics and Cluster Abundance: These are sensitive to the nature of dark energy and could show deviations from ΛCDM if w_ω(z) ≠ -1 or if ω-field perturbations have a significant effect.

7.3 Laboratory and Astrophysical Tests for IP Parameters

Precision Calorimetry: Verify $dE = \pi k_B T d\omega$ in controlled systems to constrain the fundamental IP coupling.
Complexity Wave Experiments (IP.Field): Measure κ and β² from the propagation and correlation length of local complexity fluctuations.
High-Energy/Temperature Probes: Experiments in high-temperature plasmas or high-energy particle collisions could probe the running of α_eff(T) by looking for anomalous energy-information correlations.

8. Conclusion

This paper has proposed a unified cosmological model where a single information complexity density field ω(x,t), featuring a temperature-dependent running coupling α_eff(T) and a metastable self-interaction potential V_self(ω), naturally drives three distinct cosmological phases: enhanced-coupling inflation, metastable complexity equilibrium, and relaxation-driven dark energy. This framework resolves the critical energy scale hierarchy problem between local Information Physics and cosmology and provides a physical mechanism for the timing of dark energy onset without fine-tuning.

The model’s strength lies in the “IP Consistency Test,” which demands that a single set of fundamental parameters for the ω-field {κ, β², μ³, λ₄, and parameters A, n for F(x)} must consistently explain both laboratory-scale complexity-energy phenomena and the entire observed cosmic history. This offers a pathway to a predictive, falsifiable theory where cosmic evolution is fundamentally governed by the thermodynamics and dynamics of information geometric complexity.

Future work will focus on detailed numerical solutions for the three phases, precise calculation of observational signatures (n_s, r, w_ω(z), LSS/CMB effects), and rigorous derivation of the running coupling function F(T/T_Planck) from the first principles of information theory at quantum and Planck scales. This unified ω-field cosmology represents a significant step towards understanding the universe as an information-processing system, where complexity evolution shapes the fabric of spacetime and its grandest dynamics.

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