Numerical simulation of the stochastic formalism including non-Markovianity

TL;DR

This work addresses the non-Markovian stochastic dynamics of IR modes in a de Sitter background by solving coupled IR and UV mode equations, providing a fully numerical treatment that captures memory effects arising from scale separation. The authors implement a UV/IR coarse-graining scheme and show that, when effective masses are included consistently, MSSM flat directions do not saturate but continue to diffuse along the flat direction, while non-Markovian memory effects modify stationary distributions in simple models. Through analyses of V=λφ^4 and V=μ^2φχ+λφ^4, they quantify how memory changes diffusion amplitudes and PDFs, with stronger memory evident at larger couplings and diminishing as couplings weaken. The results highlight the importance of non-Markovian dynamics for accurate predictions in stochastic inflation and open avenues for refined studies of IR phenomena in cosmology.

Abstract

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.

Figures (6)

• Figure 1: Second order moments (upper panels) and the amplitude of diffusion terms \ref{['normalized noises']} in Langevin equations (lower panels) for IR fields (left panels) and their momenta (right panels) in the MSSM system with the potential \ref{['MSSM potential']}.
• Figure 2: Second order moments (upper panels) and the amplitude of diffusion terms \ref{['normalized noises']} in Langevin equations \ref{['single phi']} and \ref{['single pi']} (lower panels) of the field for the potential $\lambda\phi^4$ with $\lambda=0.25$ (left panels) and $\lambda = 0.025$ (right panels). The number of stochastic realization are $1.0\times10^5$ and $5\times 10^4$ for the former and the latter cases. The time step is $\mathrm{d} N=0.05$.
• Figure 3: left: PDF of $\widetilde{\phi}$ for the potential $\lambda\phi^4$ with $\lambda=0.25$, obtained from the full non-Markov simulation (red) in the left panels of Fig. . right: PDF for a stationary state at $N=100$ for $\lambda=0.25$ in both the non-Markov and the Markov cases.
• Figure 4: Second order moments for $\phi$ (upper panels) and the diffusion amplitude for $\phi$\ref{['normalized noises']} in the Langevin equation \ref{['multi phi']} (lower panels) for the potential $\mu^2\phi\chi+\lambda\phi^4$ with $\mu=0.2H,\ \lambda=0.125$ (left panels), $\mu=0.2H,\ \lambda=0.005$ (middle panels), and $\mu=0.5H,\ \lambda=0.125$ (right panels). The number of stochastic realizations is $5\times10^4$, and the time step is $\mathrm{d} N=0.05$.
• Figure 5: Second order moments for $\chi$ (upper panels) and the diffusion amplitude for $\chi$\ref{['normalized noises']} in the Langevin equation \ref{['multi chi']} (lower panels) for the potential $\mu^2\phi\chi+\lambda\phi^4$ with $\mu=0.2H,\ \lambda=0.125$ (left panels), $\mu=0.2H,\ \lambda=0.005$ (middle panels), and $\mu=0.5H,\ \lambda=0.125$ (right panels). The number of stochastic realizations is $5\times10^4$, and the time step is $\mathrm{d} N=0.05$.