Linear Perturbations and Spectral Evolution

March 19, 2026

Linear Perturbations and Spectral Evolution

A. Background Evolution

The coupled field system was numerically integrated under homogeneous and isotropic conditions using the FRW metric. The dynamical variables include the scale factor a(t), the substrate field S(t), and the coupled matter-field Ψ(t).

Background equations:

H² = (8πG / 3) [ 1/2 Ṡ² + 1/2 Ψ̇² + V(S, Ψ) ]

S̈ + 3H Ṡ + β S³ = −(κ/2) Ψ²

Ψ̈ + 3H Ψ̇ + μΨ + λΨ³ = κ S Ψ

Numerical integration reveals three regimes:

Early-time potential-dominated phase: V(S,Ψ) dominates kinetic terms, producing accelerated expansion (ä > 0).
Intermediate oscillatory regime: Both fields undergo damped oscillations due to the Hubble friction term 3Hϕ̇.
Late-time relaxation: The system approaches a quasi-stable configuration near a local minimum of V(S,Ψ).

B. Linear Perturbation Dynamics

Perturbations are introduced as:

S(t,x)  = S(t)  + δS(t,x)
Ψ(t,x)  = Ψ(t)  + δΨ(t,x)

Fourier decomposition yields coupled mode equations:

δS̈_k + 3H δṠ_k + (k²/a² + 3βS²) δS_k = −κΨ δΨ_k

δΨ̈_k + 3H δΨ̇_k + (k²/a² + μ + 3λΨ² − κS) δΨ_k = κΨ δS_k

This system behaves as a pair of time-dependent, coupled oscillators with damping and nonlinear mass terms.

C. Power Spectrum Evolution

Power spectrum:

P(k,t) = |δΨ_k(t)|²

Dimensionless form:

P(k,t) = (k³ / 2π²) |δΨ_k(t)|²

D. Numerical Results

Figure 1 — Background Field Evolution

Description for paper figure:
Plot of S(t), Ψ(t), and a(t) vs time.

a(t) shows early accelerated growth.
S(t) stabilizes via cubic saturation βS³.
Ψ(t) exhibits oscillatory decay influenced by κSΨ coupling.

Figure 2 — Perturbation Mode Evolution

Description: Plot of δΨ_k(t) for a representative mode k.

Early-time growth in certain parameter regimes (instability band).
Damped oscillations as expansion increases.
Mode coupling produces phase shifts relative to uncoupled evolution.

Figure 3 — Power Spectrum P(k,t)

Description: Plot of P(k) vs time for fixed k.

Initial amplification (energy transfer from background fields).
Plateau region where modes freeze (k ≪ aH).
Late-time stabilization or decay depending on coupling strength.

E. Interpretation

1. Mode Coupling as a Structure Generator

Coupling terms κSΨ and κΨ δS_k introduce energy exchange between fields, leading to:

enhanced perturbation growth
nontrivial spectral evolution
deviation from single-field inflationary behavior

2. Effective Mass Modulation

Time-dependent effective masses:

m_S² = 3βS²
m_Ψ² = μ + 3λΨ² − κS

shifting stability regimes
transient instabilities
scale-dependent growth

3. Freeze-Out Behavior

For modes satisfying k ≪ aH:

oscillations cease
perturbations freeze
coupling introduces deviations from scale invariance

4. Nonlinear Saturation Effects

Cubic terms βS³ and λΨ³ limit growth and prevent divergence:

bounded energy density
controlled perturbation amplitudes
no singular amplification

F. Comparison to Standard Cosmology

Feature	Standard Model	This System
Fields	Single scalar	Two coupled fields
Growth	Inflation-driven	Coupling + nonlinear effects
Spectrum	Near scale-invariant	Parameter-dependent
Stability	Linearized	Nonlinear saturation

G. Key Result

Coupled nonlinear field dynamics generate structured, bounded perturbations with scale-dependent evolution.

H. Limitations

No direct calibration to observational datasets (e.g., CMB)
Single-mode simulation only
Metric perturbations not included
Parameter space not fully explored

I. Outlook

Multi-mode spectral analysis
Calculation of spectral index n_s
Inclusion of metric perturbations
Comparison with observational constraints

Conspiranon