Circuit and Krylov complexity of primordial perturbations of modified gravity in inflation

Technical Summary: Circuit and Krylov Complexity of Primordial Perturbations in Modified Gravity Inflation

Problem Statement
The microscopic origin of primordial quantum fluctuations, which seed large-scale structures and primordial black holes, remains a central puzzle in modern cosmology. While the inflationary paradigm successfully explains the scale-invariance of the primordial power spectrum, the standard canonical scalar-field model may not fully capture the complex quantum dynamics of the early universe. Recent developments in holography and quantum information theory suggest that spacetime emerges from quantum entanglement and that quantum complexity characterizes long-time dynamics. However, the specific impact of modified gravity theories, particularly $f(\phi, R)$ models, on the quantum complexity of primordial curvature perturbations has not been systematically investigated. This work addresses the gap by comparing the circuit and Krylov complexities of primordial perturbations in canonical scalar-field inflation versus $f(\phi, R)$ modified gravity.

Methodology
The authors employ a dual-diagnostic approach using quantum complexity measures:

1. Theoretical Framework: The study utilizes the quadratic action for curvature perturbations in both canonical scalar-field inflation and $f(\phi, R)$ inflation. The authors work in the Jordan frame for the modified gravity sector to keep the $f(\phi, R)$ coupling manifest in the quadratic action.
2. State Evolution: The primordial perturbations are modeled as a two-mode squeezed state generated by the coupling between $\vec{k}$ and $-\vec{k}$ momentum sectors. The evolution is governed by the squeezed strength $r_k$ and the squeezed angle $\phi_k$.
3. Circuit Complexity: Using the geometric approach (Nielsen et al.), the circuit complexity is calculated as the minimal geodesic distance in the unitary group manifold required to evolve a reference vacuum state to the target squeezed state. The complexity is derived from the wave function parameters $r_k$ and $\phi_k$.
4. Krylov Complexity: The authors utilize the Lanczos algorithm to construct an orthogonal Krylov basis. They compute:
   • Krylov Complexity ($K$): Defined as the mean pair number, $K = \sinh^2 r_k$.
   • Lanczos Coefficients ($b_n$): Derived from the recursion relations of the Liouvillian superoperator, serving as a diagnostic for operator growth and chaos.
   • Krylov Entropy ($S_K$): Calculated from the probability distribution of the operator wave function.
   • Dissipative Term ($c_n$): An effective contribution within an open-system extension to account for energy/matter exchange with the environment.
5. Numerical Simulation: The authors solve the differential equations for $r_k$ and $\phi_k$ numerically, converting the conformal time $\eta$ to $y = \log_{10} a$ to analyze the evolution across the horizon exit regime.

Key Contributions and Results

• Evolution of Squeezed Parameters: The inclusion of the $f(\phi, R)$ coupling significantly alters the evolution of the squeezed parameters.
  • Squeezed Strength ($r_k$): The $f(\phi, R)$ coupling enhances the squeezed strength relative to canonical inflation prior to horizon exit, indicating stronger quantum entanglement between the $\vec{k}$ and $-\vec{k}$ modes. However, after horizon exit, the $f(\phi, R)$ model suppresses the value of $r_k$ compared to the canonical case.
  • Squeezed Angle ($\phi_k$): Modified gravity suppresses the value of the squeezed angle prior to horizon exit, leading to a less violent spiral evolution in phase space compared to the canonical model.
• Circuit Complexity: The numerical results show that circuit complexity in the $f(\phi, R)$ framework displays a more pronounced evolution, particularly after the horizon exit. The final complexity value in the $f(\phi, R)$ model ($\sim 13$) substantially exceeds that of canonical inflation ($\sim 10$). This indicates that the modified gravity framework drives a more intricate quantum evolution, requiring a larger number of unitary gates to reach the final state. The Lyapunov exponent remains nearly identical to the canonical case, suggesting that while the complexity increases, the fundamental chaos of the dynamics is not drastically altered.
• Krylov Complexity and Chaos:
  • Complexity Growth: Since Krylov complexity is directly controlled by the mean pair number ($K = \sinh^2 r_k$), the suppression of $r_k$ in the $f(\phi, R)$ model after horizon exit leads to a smaller growth in Krylov complexity compared to the canonical case.
  • Lanczos Coefficients ($b_n$): The Lanczos coefficients, which diagnose chaos, exhibit a monotonic increase in both models. However, the chaotic behavior in $f(\phi, R)$ inflation is significantly amplified compared to the canonical case.
  • Dissipation ($c_n$): In the open-system extension, the dissipation term $c_n$ grows exponentially after horizon exit. In the $f(\phi, R)$ model, notable dissipation emerges even before horizon exit. The authors note that $c_n$ is almost equal to $b_n$, suggesting that in an open universe framework, environmental dissipation may outpace internal chaos growth, potentially leading to negative growth in Krylov complexity.
• Krylov Entropy: The entropy follows the exponential growth of the Lanczos coefficients, reinforcing the law of increasing cosmic entropy, with $f(\phi, R)$ models exhibiting greater chaos.

Significance and Claims
The paper claims that the $f(\phi, R)$ inflationary framework provides a richer and more stable theoretical environment for investigating the microscopic statistical properties of quantum fluctuations. Specifically:

• Enhanced Entanglement: The $f(\phi, R)$ coupling enhances the quantum entanglement between momentum modes prior to horizon exit, offering a promising avenue for probing the statistical properties of primordial perturbations.
• Complex Evolution: The framework reveals a highly complex evolutionary process for quantum fluctuations, characterized by pronounced physical phenomena such as transient reductions in circuit complexity followed by sharp growth.
• Structure Formation: The results suggest that $f(\phi, R)$ inflation provides a superior theoretical environment for understanding how quantum fluctuations seed large-scale structures. The rapid growth of curvature perturbations in this framework may provide favorable conditions for the formation of primordial black holes.
• Open System Dynamics: By incorporating the dissipative term $c_n$, the study highlights that treating the universe as an open system is crucial for a rigorous statistical description, potentially altering the growth trajectory of Krylov complexity.

The authors conclude that their work sheds new light on the quantum complexity of modified gravity, demonstrating that $f(\phi, R)$ models offer distinct dynamical signatures compared to canonical scalar-field inflation, particularly in the context of circuit and Krylov diagnostics.