Numerical simulation of the stochastic formalism including non-Markovianity

We numerically investigate stochastic dynamics in cosmology by solving Langevin equations for Infrared (IR) modes with stochastic noises generated by Ultraviolet (UV) modes at the coarse-graining scale. By construction, the stochastic formalism relies on the separation of scales, which requires solving the equations for UV modes on top of the evolving IR modes for all modes at every time step, leading to a non-Markovian system in general. In this paper, working on a de Sitter background, we analyze several representative models by simultaneously solving the Langevin equations for IR modes and the equations for UV modes at each time step. We demonstrate that once the effects of effective masses are treated consistently by our simulation, the flat direction in the minimal supersymmtric model (MSSM) does not saturate but instead evolves as an exactly flat direction. Furthermore, we investigate memory effects in simple two models; $V=λϕ^4$ and $V=μϕχ+ λϕ^4$, and non-Markovian contributions can lead to quantitative differences, even in stationary configurations, when compared with Markovian approximations, particularly in the strong-coupling regime.

💡 Research Summary

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The paper presents a comprehensive numerical study of the stochastic inflation formalism that fully incorporates non‑Markovian effects. Starting from the Schwinger‑Keldysh path‑integral derivation, the authors separate scalar fields into infrared (IR) modes, treated as classical stochastic variables, and ultraviolet (UV) modes, which act as a quantum environment. The interaction between the two sectors generates stochastic noise terms in the IR Langevin equations; the variance of these noises is determined by the UV mode functions evaluated at the coarse‑graining scale kσ(N)=σaH. In the usual Markovian approximation one assumes that the UV sector reacts instantaneously to the current IR background, so the noise can be computed from the instantaneous effective masses. However, the UV mode equations (Eqs. 2.10‑2.11) depend on the entire history of the IR background, making the combined system intrinsically non‑Markovian.

To capture this memory, the authors develop a fully coupled algorithm (detailed in Appendix A) that, at each e‑fold step, (i) updates the IR fields using the Langevin equations with the current noise, (ii) solves the UV mode equations for all relevant momenta on top of the current IR background, (iii) extracts the UV mode amplitudes at the moving cutoff and recomputes the noise covariance for the next step. This procedure is computationally intensive because the UV sector must be integrated for many modes at every time step, but it yields the exact stochastic dynamics without the Markovian simplifications.

The framework is applied to two classes of models. First, the authors examine a Minimal Supersymmetric Standard Model (MSSM) potential containing a flat direction coupled to massive directions. Earlier studies, based on Markovian treatments, reported that the flat direction’s variance saturates at a finite value. In contrast, the full non‑Markovian simulation shows that once the effective masses generated by the couplings are consistently accounted for, the flat direction remains exactly flat: its variance continues to grow without saturation. This result highlights the importance of correctly handling time‑dependent effective masses and the associated memory effects.

Second, the paper investigates simple single‑field and two‑field potentials, V=λϕ⁴ and V=μϕχ+λϕ⁴, respectively. By comparing Markovian approximations (fixed effective masses, analytic UV solutions) with the full non‑Markovian runs, the authors find that for weak couplings (λ, μ ≲ 0.1 H²) the two approaches agree closely. However, in the strong‑coupling regime (λ, μ ≳ H²) the non‑Markovian dynamics produce noticeable quantitative differences: the equilibrium probability distribution’s variance can be 10–30 % larger, and the tails of the distribution are altered. In the two‑field case the cross‑coupling μ introduces a time‑dependent mass matrix, which further enhances memory effects. These differences are especially relevant for calculations of rare‑event probabilities, such as primordial black‑hole formation, where the tail of the distribution dominates.

The authors also discuss the “recursive approach,” a semi‑analytic method that iteratively updates the UV noise using progressively refined IR statistics. While computationally cheaper, this method converges slowly to the true solution when strong memory effects are present, underscoring the necessity of the full simulation for accurate results in such regimes.

In summary, the study demonstrates that the stochastic inflation formalism is fundamentally non‑Markovian, and that neglecting this feature can lead to qualitatively and quantitatively incorrect predictions, particularly for models with time‑dependent effective masses, strong self‑couplings, or multiple interacting fields. The work provides a robust numerical tool for future investigations of non‑perturbative inflationary phenomena, including stochastic δN calculations, primordial black‑hole abundance estimates, and scalar‑induced gravitational wave spectra.