DSWIM:Efficient and Stable Deterministic Computation of Warm Inflation Perturbations

Technical Summary: DSWIM – Efficient and Stable Deterministic Computation of Warm Inflation Perturbations

Problem Statement
Warm inflation (WI) perturbations are driven by both quantum and thermal fluctuations, necessitating the solution of coupled stochastic differential equations. While the publicly available code SWIM implements a robust stochastic approach, it requires ensemble averaging over many realizations to achieve statistical convergence, rendering it computationally expensive for parameter inference and likelihood evaluations.

A deterministic alternative based on the Fokker-Planck approach, which evolves the correlation matrix of perturbation variables via ordinary differential equations (ODEs), offers a significant reduction in computational cost. However, the authors identify that direct deterministic evolution suffers from severe numerical ill-conditioning in many WI models. This instability arises because perturbation variables evolve over widely different scales, causing elements of the correlation matrix to span many orders of magnitude (e.g., 13 to 90 orders in tested models). This hierarchy leads to floating-point round-off errors, artificial distortions in the correction factor $G(Q)$, and eventual breakdown of the numerical evolution, particularly in the strong dissipative regime. Furthermore, previous deterministic implementations (e.g., WI2easy) exhibited systematic discrepancies with stochastic results in the moderate dissipative regime due to an incorrect treatment of correlated thermal noise.

Methodology
To address these issues, the authors introduce DSWIM (Deterministic SWIM), a new module within the SWIM framework that implements a scaling matrix framework for the deterministic evolution equations.

1. Scaling Transformation: The authors define a scaled state vector $\tilde{\Phi} = S\Phi$, where $S$ is a diagonal scaling matrix composed of powers of the Hubble parameter $H$. The scaling is physically motivated by the effective scaling behavior of the perturbation variables during inflation:

   • Metric perturbation $\psi$ (dimensionless) remains unscaled.
   • Inflaton fluctuations $\delta\phi$ and their derivatives are scaled by $H^{-1}$.
   • Radiation energy density perturbation $\delta\rho_r$ is scaled by $H^{-2}$.
   • Radiation momentum perturbation $\delta q_r$ is scaled by $H^{-1}$ based on its appearance in the perturbation equations.
     This transformation normalizes the variables, ensuring that the elements of the evolved correlation matrix remain numerically comparable throughout the integration, thereby preserving the dynamic range and preventing loss of precision.

2. Preservation of Physical Observables: The authors demonstrate that this scaling transformation leaves the comoving curvature perturbation $R$ and, consequently, the primordial scalar power spectrum $P_R$ exactly invariant. The transformed drift matrix $\tilde{A}$ and diffusion matrix $\tilde{D}$ are derived to maintain the physical consistency of the system.

3. Correction of Noise Correlations: The deterministic formalism is updated to explicitly incorporate the off-diagonal correlations in the diffusion matrix $D$. The authors show that thermal noise terms in the inflaton and radiation equations are not independent but are correlated due to energy-momentum conservation. Incorporating these non-vanishing off-diagonal terms resolves the systematic discrepancies previously observed between stochastic and deterministic codes in the moderate dissipative regime.

Key Results

• Numerical Stability: The scaled deterministic framework successfully suppresses numerical artifacts observed in unscaled implementations. For a runaway exponential potential model, the scaling transformation reduces the covariance matrix hierarchy by approximately 40–50 orders of magnitude, extending the stable evolution range of the deterministic solver by nearly an order of magnitude in the dissipation ratio $Q$ compared to unscaled methods.
• Agreement with Stochastic Methods: The results from DSWIM show excellent agreement with the stochastic SWIM implementation across weak, intermediate, and strong dissipative regimes, validating the physical accuracy of the scaled deterministic approach.
• Computational Efficiency: The deterministic approach yields substantial speedups. For representative models, DSWIM reduces the runtime for computing the correction factor $G(Q)$ by approximately two orders of magnitude compared to the stochastic SWIM and significantly outperforms the existing deterministic code WI2easy. For example, runtime for a quartic potential model dropped from ~116 seconds (stochastic) to ~1.56 seconds (deterministic).
• Resolution of Discrepancies: By correctly implementing the correlated thermal noise structure in the diffusion matrix, DSWIM eliminates the systematic deviations between stochastic and deterministic results in the moderate dissipative regime ($0.01 \lesssim Q \lesssim 10$).

Significance and Claims
The paper establishes DSWIM as a numerically robust and computationally efficient framework for computing warm inflation scalar perturbations. The authors claim that DSWIM is currently the only publicly available WI perturbation solver that provides a numerically stabilized deterministic formalism directly interfaced with the Cobaya likelihood analysis framework. This capability allows for efficient parameter inference studies of WI models that were previously computationally prohibitive due to the cost of stochastic averaging or the numerical instability of unscaled deterministic methods. The work does not propose new physical models but rather provides a necessary computational tool to accurately and efficiently explore the parameter space of existing warm inflation scenarios.