Motivated by the morphological measures in assessing the geometrical and topological properties of a generic cosmological stochastic field, we propose an extension of the weighted morphological measures, specifically the $n$th conditional moments of derivative (cmd-$n$). This criterion assigns a distinct weight to each excursion set point based on the associated field. We apply the cmd-$n$ on the Cosmic Microwave Background (CMB) to identify the cosmic string networks (CSs) through their unique Gott-Kaiser-Stebbins effect on the temperature anisotropies. We also formulate the perturbative expansion of cmd-$n$ for the weak non-Gaussian regime up to $\mathcal{O}(\sigma_0^3)$. We propose a comprehensive pipeline designed to analyze the morphological properties of string-induced CMB maps within the flat sky approximation. To evaluate the robustness of our proposed criteria, we employ string-induced high-resolution flat-sky CMB simulated patches of $7.2$ deg$^2$ size with a resolution of $0.42$ arc-minutes. Our results demonstrate that the minimum detectable value of cosmic string tension is $G\mu\gtrsim 1.9\times 10^{-7}$ when a noise-free map is analyzed with normalized cmd-$n$. Whereas for the ACT, CMB-S4, and Planck-like experiments at 95.45\% confidence level, the normalized cmd-$n$ can distinguish the CSs network for $G\mu\gtrsim2.9 \times 10^{-7}$, $G\mu\gtrsim 2.4\times 10^{-7}$ and $G\mu\gtrsim 5.8\times 10^{-7}$, respectively. The normalized cmd-$n$ exhibits a significantly enhanced capability in detecting CSs relative to the Minkowski Functionals.

翻译：受评估一般宇宙学随机场几何与拓扑性质的形态学测度启发，我们提出了加权形态学测度的扩展方案，特别是导数第n阶条件矩（cmd-$n$）。该准则根据关联场为每个逾渗集点分配不同权重。我们将cmd-$n$应用于宇宙微波背景辐射（CMB），通过宇宙弦网络在温度各向异性中独特的Gott-Kaiser-Stebbins效应来识别其存在。同时推导了弱非高斯体系下cmd-$n$直至$\mathcal{O}(\sigma_0^3)$阶的微扰展开式。我们构建了完整的分析流程，用于在平面天球近似下研究弦诱导CMB图的形态学特性。为评估所提准则的稳健性，采用弦诱导高分辨率平面天球CMB模拟天区（尺寸$7.2$平方度，分辨率$0.42$角分）进行验证。结果表明：在分析无噪声CMB图时，归一化cmd-$n$可探测的宇宙弦张力最小值为$G\mu\gtrsim 1.9\times 10^{-7}$；对于ACT、CMB-S4及类Planck实验，在95.45\%置信水平下，归一化cmd-$n$可分别探测$G\mu\gtrsim2.9 \times 10^{-7}$、$G\mu\gtrsim 2.4\times 10^{-7}$和$G\mu\gtrsim 5.8\times 10^{-7}$的宇宙弦网络。相较于闵可夫斯基泛函，归一化cmd-$n$展现出显著增强的宇宙弦探测能力。