标题： 标度相关的霍恩德斯基本理论中的单圈核 显示英文标题

标题： One-loop kernels in scale-dependent Horndeski theory

摘要： 我们研究了具有尺度相关扰动增长的理论中宇宙扰动的非线性演化，首先从一般情况开始，然后聚焦于Horndeski引力。在标准扰动理论框架下，我们推导出二阶和三阶核，并表明它们完全由两个有效函数 $h_1$ 和 $h_c$ 确定，这些函数参数化了与广义相对论的偏差。利用Wronskian方法，我们获得了非线性增长函数的解，并给出了由此产生的核的显式表达式，包括偏置和红移空间畸变。我们证明这些核完全依赖于线性增长模式：一旦计算出该模式，核就可以解析地表示为一个时间积分。我们的方法为评估尺度相关理论中的单圈星系功率谱提供了一个物理上有意义的框架，适用于预测和实际数据分析。 显示英文摘要

摘要： We investigate the nonlinear evolution of cosmological perturbations in theories with scale-dependent perturbation growth, first in general and then focusing on Horndeski gravity. Within the framework of standard perturbation theory, we derive the second- and third-order kernels and show that they are fully determined by two effective functions, $h_1$ and $h_c$, which parametrize deviations from general relativity. Using the Wronskian method, we obtain solutions for the nonlinear growth functions and present explicit expressions for the resulting kernels, including bias and redshift space distortions. We show that the kernels are entirely dependent on the linear growing mode: once this is calculated, the kernels are analytic up to a time integral. Our approach provides a physically motivated framework for evaluating the one-loop galaxy power spectrum in scale-dependent theories, suitable for the forecasts and actual data analysis.