Quantum-information diagnostics of cosmological perturbations with nontrivial sound speed in inflation

Abstract: In this work, we systematically investigate the quantum-information diagnostics of cosmological perturbations with a nontrivial sound speed, utilizing a normalized open two-mode squeezed-state framework. Rather than introducing new observables, our analysis focuses on how a modified sound speed dynamically reshapes the Schrödinger evolution of the squeezing parameters ($r_k$ and $φ_k$). We demonstrate how these dynamical changes are inherited by the reduced density matrix of the observable sector. By employing a sound-speed-resonance parametrization, we derive and evaluate the purity, von Neumann entropy, Rényi entropies, and logarithmic negativity. To overcome the intrinsic multiscale stiffness of the post-inflationary equations, we introduce a bounded variable $x = \tanh r_k$ as a partial regularization, which enables reliable numerical simulations exclusively within the inflationary regime. Our numerical results reveal that a nontrivial sound speed significantly suppresses the purity of the reduced state, indicating enhanced effective mixedness. Simultaneously, it strongly amplifies and modulates both the entropic and entanglement diagnostics. More precisely, a nontrivial sound speed postpones the onset of classicality by modulating the decoherence process. Ultimately, we show that a nontrivial sound speed leaves distinct and identifiable quantum-information signatures within the entanglement structure of the early universe.

• The paper demonstrates that nontrivial sound speed modifies the quantum-information diagnostics of cosmological perturbations by dynamically altering squeezing parameters.
• The analysis utilizes the OTMSS formalism and a sound-speed-resonance parametrization to quantitatively track oscillatory modulations in purity, von Neumann entropy, and logarithmic negativity.
• Numerical simulations reveal that SSR delays classicality onset by enhancing decoherence and entanglement, suggesting new avenues for probing early-universe inflationary dynamics.

Quantum-Information Diagnostics of Inflationary Cosmological Perturbations with Nontrivial Sound Speed

Motivation and Framework

The quantum mechanical origin of primordial cosmological perturbations remains a pivotal topic in modern cosmology. The paper addresses how quantum-information diagnostics, including purity, von Neumann entropy, R\'enyi entropies, and logarithmic negativity, are influenced by inflationary models with nontrivial sound speed ($c_s \neq 1$). Focusing on the open two-mode squeezed-state (OTMSS) formalism, the analysis circumvents the modification of observable definitions and instead tracks the dynamical imprint of sound speed through the evolution of squeezing parameters, $r_k$ (amplitude) and $\phi_k$ (phase), as dictated by the inflationary Hamiltonian.

A sound-speed-resonance (SSR) parametrization is employed:

$c_s^2(\eta) = 1 - 2\xi [1 - \cos(k\eta)]$

with $\xi$ modulating oscillatory deviations from canonical $c_s=1$, thereby directly altering the effective frequency in the Mukhanov–Sasaki equation and ultimately the quantum state evolution. The observable sector is treated via the reduced density matrix (RDM) after tracing out environmental degrees of freedom, enabling rigorous information-theoretic diagnostics of subsystem mixedness and entanglement.

Squeezing Dynamics and Sound-Speed Resonance

The inflationary dynamics are formulated within a spatially flat FLRW metric, relating the scale factor $a(\eta)$ to conformal time and utilizing the SSR ansatz for sound-speed oscillations. The quadratic action and subsequent Hamiltonian elucidate how $c_s$ enters explicitly:

$$H = \frac{1}{2} \int d^3x \left[ \pi^2 + c_s^2 (\partial_i f)^2 + \frac{z'}{z} (\pi f + f\pi) + a^2V_{\phi\phi} f^2 \right]$$

where both background-driven squeezing and sound-speed contributions are transparent. Fourier expansion and mode decomposition allow the explicit construction of the OTMSS, with normalization controlled via dissipation coefficients $u_1$ and $r_k$0. The squeezing amplitude $r_k$1 and phase $r_k$2 evolve through coupled, stiff Schrödinger equations, regularized by the bounded variable $r_k$3 to ensure computational stability within the inflationary regime.

Within $r_k$4, numerical simulations demonstrate that nontrivial sound speed ($r_k$5) induces pronounced oscillatory modulation of $r_k$6 and $r_k$7, confirming significant dynamical sensitivity.

Figure 1: The numerical results of $r_k$8 in terms of $r_k$9, demonstrating SSR-induced oscillatory modulation for various $\phi_k$0.

Figure 2: The numerical results of $\phi_k$1 in terms of $\phi_k$2, highlighting the SSR-driven suppression and modulation compared to the canonical case.

Quantum-Information Diagnostics: Construction and Interpretation

Tracing out one mode yields the RDM, diagonal in the occupation-number basis, with eigenvalues parameterized by the squeezing variables and open-system coefficients. The information-theoretic observables are computed as follows:

• Purity: $\phi_k$3, sensitive to subsystem mixedness.
• Von Neumann entropy: $\phi_k$4 quantifies uncertainty and entropy production.
• R\'enyi entropies: $\phi_k$5 generalize the entropy, with $\phi_k$6 and $\phi_k$7 serving as robust numerical bounds.
• Logarithmic negativity: $\phi_k$8 diagnoses genuine bipartite entanglement, derived from the partial transpose spectrum.

Numerical results indicate that SSR leads to substantial suppression of purity (enhanced mixedness) and amplification/modulation of entropic measures and negativity. This is not the result of altered algebraic definitions, but the inherited dynamical trajectory of the quantum state caused by the Hamiltonian's dependence on $\phi_k$9.

Figure 3: The numerical results of Purity in terms of $c_s^2(\eta) = 1 - 2\xi [1 - \cos(k\eta)]$0, showing SSR-induced suppression and oscillatory behavior.

Figure 4: The numerical results of R\'enyi Entropy and Von Neumann entropy, presenting enhanced entropy production and modulation with increasing $c_s^2(\eta) = 1 - 2\xi [1 - \cos(k\eta)]$1.

Figure 5: The numerical results of Logarithmic negativity, exhibiting amplified entanglement signatures under SSR.

Implications and Future Directions

The findings substantiate that a nontrivial sound speed during inflation postpones classicality onset by modulating decoherence and amplifying subsystem mixedness and entanglement structure. SSR leaves distinct quantum-information fingerprints in the early universe, with observable departures from canonical single-field models as evidenced by oscillatory suppression in purity and enhancement in entropy and negativity.

Numerical stiffness restricts analysis to the inflationary regime, but future work aims to extend diagnostics to RD and MD epochs via lattice methods. The paradigm and regularization strategy promise generalization to multi-field, $c_s^2(\eta) = 1 - 2\xi [1 - \cos(k\eta)]$2, and non-inertial cosmological scenarios.

Conclusion

The systematic study delivers strong numerical evidence that nontrivial sound speed in inflation fundamentally alters quantum-information diagnostics of cosmological perturbations. The SSR-induced modifications are dynamically inherited, not algebraically imposed, affecting the decoherence and entanglement structure of primordial fluctuations. The paper offers a robust formalism to extract quantum-information signatures, guiding both theoretical interpretation and future empirical investigation of inflationary models. Potential advancements lie in rigorous covariance-matrix-based decoherence quantification, cross-epoch numerical stabilization, and generalization to broader gravitational and field-theoretic settings.