标题： 通过蒙特卡罗模拟和最小二乘曲线拟合计算随机暴胀中的功率谱 显示英文标题

标题： Calculating the power spectrum in stochastic inflation by Monte Carlo simulation and least squares curve fitting

摘要： 随机-$\delta \mathcal{N}$形式广泛用于研究量子扩散主导背景动力学的暴胀模型，导致了原初黑洞产生等有趣现象。 在该形式中计算曲率扰动谱 $\mathcal{P}_\zeta(k)$的数值方法中，基于蒙特卡洛模拟的方法被认为是一个有前景的选择，尤其是在多场情况下。 在此方法中，我们从初始点生成许多暴胀子路径到暴胀结束，从路径中获得 $\delta N$的统计信息，然后估计 $\mathcal{P}_\zeta(k)$。 然而，这种方法涉及嵌套的蒙特卡洛模拟，这需要在对应于感兴趣尺度 $k$的点上，从每个主路径生成许多分支路径，导致计算成本很高。 在本文中，我们提出了一种新的基于蒙特卡洛的方法，该方法利用最小二乘拟合，引入了两种新特性以减少计算成本。 首先，我们设计了一个关键统计量 $\langle \delta\mathcal{N}_{\mathbf{X}}^2\rangle$的简单估计器，即在分支点条件下 $\delta \mathcal{N}$的方差，以避免嵌套路径生成。 其次，通过将参数函数拟合到估计量的采样值，我们不仅获得了一个关于$\mathcal{P}_\zeta(k)$的单个$k$值的估计，还获得了在感兴趣的$k$范围内$\mathcal{P}_\zeta(k)$的近似函数。 我们还对具体的通胀模型进行了数值演示，这些演示展示了我们方法的实用性。 显示英文摘要

摘要： The stochastic-$\delta \mathcal{N}$ formalism is widely used to study inflation models in which the quantum diffusion of inflatons dominates the background dynamics, leading to interesting phenomena such as the production of primordial black holes. Among numerical approaches to calculate the curvature perturbation spectrum $\mathcal{P}_\zeta(k)$ in this formalism, the Monte Carlo simulation-based approach has been proposed as a promising choice, especially in multifield cases. In this approach, we generate many paths of inflatons from the initial points to the end of inflation, obtain statistics of $\delta N$ from the paths, and then estimate $\mathcal{P}_\zeta(k)$. However, this method involves a nested Monte Carlo simulation, which requires generating many branch paths from each trunk path at the point corresponding to the scale $k$ of interest, resulting in a high computational cost. In this paper, we propose a new Monte Carlo-based approach that utilizes least squares fitting, introducing two novel features for reducing computational cost. First, we devise a simple estimator of a key statistic $\langle \delta\mathcal{N}_{\mathbf{X}}^2\rangle$, the variance of $\delta \mathcal{N}$ conditioned on the branching point, to avoid nesting path generation. Second, via least squares fitting of a parametric function to the sampled values of the estimator, we obtain not just an estimate of $\mathcal{P}_\zeta(k)$ for a single value of $k$ but an approximating function of $\mathcal{P}_\zeta(k)$ over a range of $k$ of interest. We also conduct numerical demonstrations for concrete inflation models, which show the usefulness of our method.